Triple integral calculator spherical coordinates

Polar Rectangular Regions of Integration. When we defined the double integral for a continuous function in rectangular coordinates—say, \(g\) over a region \(R\) in the \(xy\)-plane—we divided \(R\) into subrectangles with sides parallel to the coordinate axes.

Objectives:9. Use iterated integrals to evaluate triple integrals in spherical coordinates.10. Find volumes using iterated integrals in spherical coordinates.Mar 3, 2024 · scssCopy code. ∫∫∫ ρ²sin(φ) dρ dφ dθ. with ρ bounds from 0 to R, φ from 0 to π, and θ from 0 to 2π. Evaluating this integral yields the volume of a sphere, 4/3πR³, demonstrating the calculator’s utility in practical applications.Lecture 17: Triple integrals IfRRR f(x,y,z) is a function and E is a bounded solid region in R3, then E f(x,y,z) dxdydz is defined as the n → ∞ limit of the Riemann sum 1 n3 X (i/n,j/n,k/n)∈E f(i n, j n, k n) . As in two dimensions, triple integrals can be evaluated by iterated 1D integral computations. Here is a simple example:2. Set up the coordinate-independent integral. We are dealing with volume integrals in three dimensions, so we will use a volume differential and integrate over a volume. Most of the time, you will have an expression in the integrand. If so, make sure that it is in spherical coordinates. 3. Set up the volume element.Evaluating Triple Integrals with Spherical Coordinates. Formula 3 says that we convert a triple integral from rectangular coordinates to spherical coordinates by writing. x = ρsin φcos θ. y = ρsin φsin θ. z = ρcos φ. using the appropriate limits of integration, and replacing . dv. by ρ. 2. sin φ. d. ρ. d. θ. d. φ.Objectives:9. Use iterated integrals to evaluate triple integrals in spherical coordinates.10. Find volumes using iterated integrals in spherical coordinates.

Overall, the resulting iterated integral in cartesian coordinates and the spherical coordinates is equal to $\frac{\pi}{2}$. I want to know now if my understanding about the conversion is correct. Is there a visual representation of this integral to fully understand on how triple integral in spherical coordinates works?11.8.4 Triple Integrals in Spherical Coordinates. 11.8.5 Summary. 11.8.6 Exercises. 11.9 Change of Variables. 11.9.1 Change of Variables in Polar Coordinates. ... Note well: in some problems you may be able to use a double rather than a triple integral, and polar coordinates may be helpful in some cases.Spherical coordinates to calculate triple integral. 1. Find the range of surface integral using spherical coordinates. 0. Tough Moment of Inertia Problem About a Super Thin Spherical Shell Using Spherical Coordinates. 4. ... Stealth In Space Calculator What is the difference in the usage of the verbs "lernen" and "studieren"? ...A Triple Integral Calculator is an online tool used to compute the triple integral of three-dimensional space and the spherical directions that determine the location of a given point in three-dimensional (3D) space depending on the distance ρ from the origin and two points $\theta$ and $\phi$.Open triples integrals numerical - solve triple integrals step-by-step

Triple integral of function of three variables in rectangular (Cartesian) coordinates. อินทิกรัลสามชั้นในพิกัดฉาก. Get the free "Triple Integral in Cartesian Coordinates" widget for your website, blog, Wordpress, Blogger, or iGoogle. Find more Mathematics widgets in Wolfram|Alpha.To convert from rectangular coordinates (x, y, z) to spherical coordinates (ρ, θ, φ), use the following relations: ρ = sqrt (x² + y² + z²), θ = atan2 (y, x), φ = acos (z / …The Integral Calculator solves an indefinite integral of a function. You can also get a better visual and understanding of the function and area under the curve using our graphing tool. Integration by parts formula: ? u d v = u v-? v d u. Step 2: Click the blue arrow to submit. Choose "Evaluate the Integral" from the topic selector and click to ...Explore math with our beautiful, free online graphing calculator. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. Spherical Coordinate System | DesmosSection 3.7 Triple Integrals in Spherical Coordinates Subsection 3.7.1 Spherical Coordinates In the event that we wish to compute, for example, the mass of an object that is invariant under rotations about the origin, it is advantageous to use another generalization of polar coordinates to three dimensions.Question: Convert the following triple integral to spherical coordinates. SETUP ONLY, DO NOT EVALUATE. integral_-1^1 integral_0^Squareroot 1 - x^2 integral_0^Squareroot 1 - x^2 - y^2 e^(x^2 + y^2 + z^2)^3/2 dz dy dx

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Use triple integrals to calculate the volume. Consider each part of the balloon separately. (Consider using spherical coordinates for the top part and cylindrical coordinates for the bottom part.) Verify the answer using the formulas for the volume of a sphere, V = 4 3 π r 3, V = 4 3 π r 3, and for the volume of a cone, V = 1 3 π r 2 h. V ...Use spherical coordinates to calculate the triple integral of f ( x, y, z) = x 2 + y 2 + z 2 over the region. 1 ≤ x 2 + y 2 + z 2 ≤ 4 9. ( Give an exact answer. Use symbolic notation and fractions where needed.) ∭ W f ( x, y, z) d V =. There are 2 steps to solve this one. Created by Chegg.Figure \(\PageIndex{4}\): Differential of volume in spherical coordinates (CC BY-NC-SA; Marcia Levitus) We will exemplify the use of triple integrals in spherical coordinates with some problems from quantum mechanics. We already introduced the Schrödinger equation, and even solved it for a simple system in Section 5.4. We also mentioned that ...Use spherical coordinates to evaluate the triple integral E x^2 + y^2 + z^2 dV , where E is the ball: x^2 + y^2 + z^2 49. Show transcribed image text. Here's the best way to solve it. 100% (6 ratings) View the full answer. Previous question Next question. Transcribed image text:

Title Triple integrals in cylindrical and spherical coordinates ection Stewart 15.7 15.8. arm up Spse E is bounded by y 4 x2 z and y o Express the volume of E as 6 different iterated integrals ... Triple integrals w polar coordinates theorem Spse E x y Z x y in D and a title ZE lez x y w D r 011 210113 and h O e r e halos SSS F x y z du SS Jui ...Oct 26, 2022 · Objectives:9. Use iterated integrals to evaluate triple integrals in spherical coordinates.10. Find volumes using iterated integrals in spherical coordinates.To evaluate the triple integral of f (rho, theta, phi) = cos (phi) over the given region in spherical coordinates, we need to use the correct setup for the integral. The integral should be set up as follows: ∫∫∫ cos (phi) * rho^2 * sin (phi) d (rho) d (phi) d (theta) The limits of integration are: - For rho: 3 to 7.En esta sección se define la integral triple de una función f(x,y,z) de tres variables sobre una región en el espacio. Se muestra cómo calcular la integral triple usando coordenadas cartesianas, cilíndricas y esféricas, y cómo aplicarla a problemas de volumen, masa, centro de masa y momento de inercia. También se explora la relación entre la integral triple y la divergencia de un ...Added May 7, 2021 by Rss in Mathematics. Triple Integrals - Spherical Coordinates. Send feedback | Visit Wolfram|Alpha. Get the free "Triple Integrals - Spherical Coordinates" widget for your website, blog, Wordpress, Blogger, or iGoogle. Find more Mathematics widgets in Wolfram|Alpha.Use spherical coordinates to calculate the triple integral of f (x, y, z) = x 2 + y 2 + z 2 1 over the region 5 ≤ x 2 + y 2 + z 2 ≤ 16. (Use symbolic notation and fractions where needed.) ∭ w x 2 + y 2 + z 2 1 d V Use spherical coordinates to calculate the triple integral of f (x, y, z) = x 2 + y 2 + z 2 over the region x 2 + y 2 + z 2 ...Use spherical coordinates to evaluate the integral \[ I=\iiint_D z\ \mathrm{d}V onumber \] where \(D\) is the solid enclosed by the cone \(z = \sqrt{x^2 + y^2}\) and the sphere \(x^2 + y^2 + z^2 = 4\text{.}\)In Rectangular Coordinates, the volume element, " dV " is a parallelopiped with sides: " dx ", " dy ", and " dz ". Accordingly, its volume is the product of its three sides, namely dV = dx ⋅ dy ⋅ dz . The parallelopiped is the simplest 3-dimensional solid. That it is also the basic infinitesimal volume element in the simplest coordinate ...

The spherical 3d integral calculator is a specialized mathematical tool to evaluate triple integrals expressed in spherical coordinates. Spherical coordinates are often …

Triple Integrals in Spherical Coordinates Proposition (Triple Integral in Spherical Coordinates) Let f(x;y;z) 2C(E) s.t. E ˆR3 is a closed & bounded solid . Then: ZZZ E f dV SPH= Z Largest -val in E Smallest -val in E Z Largest ˚-val in E Smallest ˚-val in E Z Outside BS of E Inside BS of E fˆ2 sin˚dˆd˚d = ZZZ E f(ˆsin˚cos ;ˆsin˚sin ...Question: (3.5 points) Set up, but do NOT evaluate, a triple integral in spherical coordinates representing the volume of the solid S above xy-plane and entrapped inside the sphere x2+y2+z2=16 and below the cone z=x2+y2. Show transcribed image text. There’s just one step to solve this.Both the integrand and the nature of B B suggest a change to spherical coordinates. As we know, the volume elements for Cartesian and spherical coordinates are related by dxdydz =ρ2 sin(φ)dρdφdθ d x d y d z = ρ 2 sin. ⁡. ( φ) d ρ d φ d θ, so the original triple integral is equal to the following iterated integral in spherical ...Example 14.5.1: Evaluating a Triple Integral. Evaluate the triple integral ∫z = 1 z = 0∫y = 4 y = 2∫x = 5 x = − 1(x + yz2)dxdydz. Solution. The order of integration is specified in the problem, so integrate with respect to x first, then y, and then z. ∫z = 1 z = 0∫y = 4 y = 2∫x = 5 x = − 1(x + yz2)dxdydz = ∫z = 1 z = 0∫y = 4 ...A triple integral in spherical coordinates calculator is a specialized tool designed to compute the volume of a three-dimensional object by integrating over a region defined in spherical coordinates.Our expert help has broken down your problem into an easy-to-learn solution you can count on. Question: Express the triple integral below in spherical coordinates and evaluate. ∭E5xdV where E is the portion of the ball x2+y2+z2≤1 that lies in the first octant. Your Answer: Answer Question 2 (13 points) Find the volume of the solid enclosed ...Shady operators are trying to game Facebook. Here's how to use the platforms features to spot them. “Coordinated Inauthentic Behavior,” a phrase coined by Facebook, is the use of m...A triple integral in spherical coordinates calculator is a specialized tool designed to compute the volume of a three-dimensional object by integrating over a region defined in spherical coordinates.Step 1. The given integral needs to be evaluated using spherical coordinates. Use spherical coordinates to find the triple integral. (Give an exact answer. Use symbolic notation and fractions where needed.) ∫ −66 ∫ − 36−y236−y2 ∫ 66+ 36−x2−y2 ydzdxdy = ∫ −66 ∫ − 36−y236−y2 ∫ 66+ 36−x2−y2 Incorrect ρ Find the ...Embed this widget ». Added May 7, 2015 by panda.panda in Mathematics. Triple integration in spherical coordinates. Send feedback | Visit Wolfram|Alpha. Get the free "Spherical Integral Calculator" widget for your website, blog, Wordpress, Blogger, or iGoogle. Find more Mathematics widgets in Wolfram|Alpha.

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You can do it geometrically, by drawing right triangles (for the first cone, you have a z = r z = r, so it's an isosceles right triangle, and ϕ = π/4 ϕ = π / 4. Alternatively, put spherical coordinates into the equation and you'll get ρ cos ϕ = ρ sin ϕ ρ cos. ϕ, so ϕ = π/4 ϕ = π / 4. You can work on the other one.Step 1. using spherical coordinates, over the region x 2 + y 2 + z 2 ≤ 8 z. Le... Use spherical coordinates to calculate the triple integral of f (x,y,z)= x2 +y2+z2 over the region x2 +y2+z2 ≤8z. (Use symbolic notation and fractions where needed.) ∭ W x2+y2+z2dV = Incorrect.Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this siteThe Jacobian is the prefactor of dS d S when changing coordinates. Typically the Jacobian is memorised for popular coordinate systems, so you would just look up that dS =nr2 sin θdϕdθ d S = n r 2 sin. ⁡. θ d ϕ d θ on the surface of a sphere, in spherical coordinates. Here n =ar n = a r is the unit normal (sin θ cos ϕ, sin θ sin ϕ ...Question: Given the triple integral in spherical coordinates, ∫02π∫03π∫02ρ2sinϕdρdϕdθ, 1. Draw the solid represented by this triple integral. 2. Evaluate this integral giving the exact value. Show transcribed image text. There are 2 steps to solve this one. Expert-verified.View the full answer. Previous question Next question. Transcribed image text: (1 pt) Use spherical coordinates to calculate the triple integral of (Use symbolic notation and fractions where needed.) х у z) = x2 2 + Z2 over the region x2 + y2 + Z2 72 help (fractions) Preview Answers Submit Answers.How is trigonometric substitution done with a triple integral? For instance, $$ 8 \int_0^r \int_0^{\sqrt{r^2-x^2}} \int_0^{\sqrt{r^2-x^2-y^2}} (1) dz dy dx $$ Here the limits have been chosen to ... $\begingroup$ I understand the switch to spherical coordinates, the question is geared toward multi-variate trig subs. $\endgroup$ - Jon. Jun 1 ...vert from spherical to rectangular coordi-nates, we use the equations x = ρsin φcosθ y θ and z ρ2 = x2 + y2 + z2 Evaluating Triple integrals with spherical coordinates In the spherical coordinate system the counterpart of a rectangular box is a spherical wedge E= {(ρ,θ,φ)|a≤ρ≤b, α≤θ≤β, c≤φ≤d} where a≥0 and β−α≤ ...Triple integrals and spherical coordinates Hello ladies and gentlemen, I have a mathematical problem where I need to determine the hypervolume of f(x,y,z) = (x^2+y^2) over the domain E located inside the sphere x^2+y^2+(z-3)^2=9 and above the half-cone z=2-sqrt(x^2+y^2). ….

Evaluate, in spherical coordinates, the triple integral of f (ρ, θ, ϕ) = cos ϕ, over the region 0 ≤ θ ≤ 2 π, π /3 ≤ ϕ ≤ π /2, 3 ≤ ρ ≤ 7. Not the question you're looking for? Post any question and get expert help quickly.Spherical coordinates, also called spherical polar coordinates (Walton 1967, Arfken 1985), are a system of curvilinear coordinates that are natural for describing positions on a sphere or spheroid. Define theta to be the azimuthal angle in the xy-plane from the x-axis with 0<=theta<2pi (denoted lambda when referred to as the longitude), phi to be the polar angle (also known as the zenith angle ...Explore math with our beautiful, free online graphing calculator. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. Spherical Coordinate System | DesmosSet up a triple integral over this region with a function \ (f (r, \theta, z)\) in cylindrical coordinates. Figure \ (\PageIndex {3}\): Setting up a triple integral in cylindrical coordinates over a cylindrical region. Solution. First, identify that the equation for the sphere is \ (r^2 + z^2 = 16\).To evaluate the triple integral of f (rho, theta, phi) = cos (phi) over the given region in spherical coordinates, we need to use the correct setup for the integral. The integral should be set up as follows: ∫∫∫ cos (phi) * rho^2 * sin (phi) d (rho) d (phi) d (theta) The limits of integration are: - For rho: 3 to 7.In today’s digital age, Excel files have become an integral part of our professional lives. They help us organize data, create spreadsheets, and perform complex calculations with e...Mar 9, 2019 ... In this video we use cylindrical coordinates and a triple integral to find the volume of a solid. (Specifically the solid bounded by z = x^2 ...Free triple integrals calculator - solve triple integrals step-by-step ... Polynomials Rationales Functions Arithmetic & Comp. Coordinate Geometry Plane Geometry ... Triple integral calculator spherical coordinates, Use spherical coordinates to find the triple integral. (Give an exact answer. Use symbolic notation and fractions where needed.) ∫ 0 4 ∫ 0 16 − y 2 ∫ 0 16 − x 2 − y 2 e (x 2 + y 2 + z 2) 3/2 d z d x d y = 3 4 π (e 64 − 1) Incorrect, Free triplet integrals calculator - solve triple integrators step-by-step ... Polynomials Rationales Functions Arithmetic & Comp. Coordinate Geometry Plane Geometry ..., I'm currently learning how to calculate the volume of a 3D surface expressed in spherical coordinates using triple integrals. There was this exercice (from here ) which asked me to find the volume of the region described by those two equations:, Use Calculator to Convert Rectangular to Spherical Coordinates. 1 - Enter x x, y y and z z and press the button "Convert". You may also change the number of decimal places as needed; it has to be a positive integer. The angles θ θ and ϕ ϕ are given in radians and degrees. (x,y,z) ( x, y, z) = (. 1., Figure 4.5.3: Setting up a triple integral in cylindrical coordinates over a cylindrical region. Solution. First, identify that the equation for the sphere is r2 + z2 = 16. We can see that the limits for z are from 0 to z = √16 − r2. Then the limits for r are from 0 to r = 2sinθ., Question: Evaluate the integral below by changing to spherical coordinates. Triple integral (x^2z+Y^2z+z^3)dz dx dY Find the volume of the solid that lies within the sphere x^2 + y^2 + z^2 = 9, above the xy-plane, and below the cone z = root x^2 + y^2. Show transcribed image text. There are 4 steps to solve this one. Expert-verified., The spherical coordinates are often used to perform volume calculations via a triple integration by changing variables: ∭ f(x,y,z) dx dy dz= ∭ f(ρcos(θ)sin(φ),ρsin(θ)sin(φ), ρcos(φ))ρ2sin(φ) dρ dθ dφ ∭ f ( x, y, z) d x d y d z = ∭ f ( ρ cos. ⁡. ( θ) sin. ⁡. ( φ), ρ sin. ⁡., We present an example of calculating a triple integral using spherical coordinates.http://www.michael-penn.nethttp://www.randolphcollege.edu/mathematics/, The process of changing variables transforms the integral in terms of the variables (x, y, z) over the dome W to an integral in terms of the variables (ρ, θ, ϕ) over the region W ∗. Since the function f(x, y, z) is defined in terms of (x, y, z), we cannot simply integrate f over the box W ∗. Instead, we must first compose f with the ..., In the spherical coordinate system, we again use an ordered triple to describe the location of a point in space. In this case, the triple describes one distance and two angles. Spherical coordinates make it simple to describe a sphere, just as cylindrical coordinates make it easy to describe a cylinder., Step 1. Evaluate the following integral in spherical coordinates SJC e- (x2 + y2 +22) 3/2 dV;D is a ball of radius 7 Set up the triple integral using spherical coordinates that should be used to evaluate the given integral as efficiently as possible. Use increasing limits of integration SSS dp dp do DO 0 0 Evaluate the integral dV = D (Type an ..., Use triple integrals to calculate the volume. Consider each part of the balloon separately. (Consider using spherical coordinates for the top part and cylindrical coordinates for the bottom part.) Verify the answer using the formulas for the volume of a sphere, V = 4 3 π r 3, V = 4 3 π r 3, and for the volume of a cone, V = 1 3 π r 2 h. V ..., We present an example of calculating a triple integral using spherical coordinates.http://www.michael-penn.nethttp://www.randolphcollege.edu/mathematics/, To evaluate the triple integral of f (rho, theta, phi) = cos (phi) over the given region in spherical coordinates, we need to use the correct setup for the integral. The integral should be set up as follows: ∫∫∫ cos (phi) * rho^2 * sin (phi) d (rho) d (phi) d (theta) The limits of integration are: - For rho: 3 to 7., triple integral calculator cylindrical. Natural Language; Math Input; Extended Keyboard Examples Upload Random. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. For math, science, nutrition, history, geography, engineering, mathematics, linguistics, sports, finance, music…, How to compute triple integral in spherical coordinates. 3. ... Can I solve this integral without spherical coordinates? 0. Bounds of integration in spherical coordinates. 2. Doubt about the use of partial derivatives: what's the solution? 2. Evaluate the triple integral $\iiint\limits_E\frac{yz\,dx\,dy\,dz}{x^2+y^2+z^2}$ using spherical ..., Calculus questions and answers. Express the triple integral below in spherical coordinates. tripleintegral_E -2 x e^x^2 + y^2 + z^2 dV where E is the portion of the ball x^2 + y^2 + z^2 lessthanorequalto 4 that lies in the first octant. tripleintegral_E -2 x e^x^2 _ y^2 + z^2 dV = integral^theta_2_theta_1 integral^phi_2_phi_1 integral^rho_2_rho ..., Welcome to our Triple Integral Calculator Cylindrical and Triple Integral Calculator spherical, a powerful tool designed to assist you in solving triple integrals quickly and accurately. Whether you are a student, a researcher, or a professional in the field of mathematics or physics, this tool can streamline your computations, saving you ..., Figure 4.6.3: Setting up a triple integral in cylindrical coordinates over a cylindrical region. Solution. First, identify that the equation for the sphere is r2 + z2 = 16. We can see that the limits for z are from 0 to z = √16 − r2. Then the limits for r are from 0 to r = 2sinθ., We can use triple integrals and spherical coordinates to solve for the volume of a solid sphere. To convert from rectangular coordinates to spherical coordinates, we use a set of spherical conversion formulas. About Pricing Login GET STARTED About Pricing Login. Step-by-step math courses covering Pre-Algebra through Calculus 3. ..., Cuba is tripping the size of its port terminal as it prepares for a record number of tourists. As Cuba gears up for increased tourism numbers, the communist country is bulking up i..., Our expert help has broken down your problem into an easy-to-learn solution you can count on. Question: Use spherical coordinates to calculate the triple integral of f (x,y,z)=x2+y2+z2 over the region x2+y2+z2≤2z. (Use symbolic notation and fractions where needed.) ∭Wx2+y2+z2dV= [. There are 3 steps to solve this one., This trio of berry flavors with a hint of nuttiness will brighten your day. It’s a super simple shake to make and chock full of powerhouse nutrients. Average Rating: This trio of b..., Triple integrals in spherical coordinates. Added Apr 22, 2015 by MaxArias in Mathematics. Give it whatever function you want expressed in spherical coordinates, choose the order of integration and choose the limits., Computers have revolutionized the way we work and play, making tasks quicker and more efficient. One tool that has become an integral part of our digital lives is the calculator. T..., Therefore, a triple integral in rectangular coordinates can be rewritten in terms of spherical coordinates: \iiint_D f (x,y,z)\ dV = \iiint_D f (\rho, \phi, \theta)\ \rho^2 \sin \phi\ d\rho\ d\phi\ d\theta ∭ D f (x,y, z) dV = ∭ D f (ρ, ϕ,θ) ρ2 sinϕ dρ dϕ dθ. We'll tend to use spherical coordinates when we encounter a triple integral ..., Example 2.6.6: Setting up a Triple Integral in Spherical Coordinates. Set up an integral for the volume of the region bounded by the cone z = √3(x2 + y2) and the hemisphere z = √4 − x2 − y2 (see the figure below). Figure 2.6.9: A region bounded below by a cone and above by a hemisphere. Solution., Lecture 18: Spherical Coordinates Cylindrical coordinates are coordinates in space in polar coordinates are used in the xy-plane and where the z-coordinate is untouched. A surface of revolution x2 + y2 = g(z)2 can be described in cylindrical coordinates as r = g(z). The coordinate change transformation, vert from spherical to rectangular coordi-nates, we use the equations x = ρsin φcosθ y θ and z ρ2 = x2 + y2 + z2 Evaluating Triple integrals with spherical coordinates In the spherical coordinate system the counterpart of a rectangular box is a spherical wedge E= {(ρ,θ,φ)|a≤ρ≤b, α≤θ≤β, c≤φ≤d} where a≥0 and β−α≤ ..., In order to use the triple integral average value formula, we'll have find the volume of the object, plus the domain of x, y, and z so that we can set limits of integration, turn the triple integral into an iterated integral, and replace dV with dzdydx. ... ??? and three sides lying in the coordinate planes.???f(x,y,z)=3xyz^2??? We'll start ..., Double integrals can also be used to find the mass of a thin plate given density (and you can find center of mass, too). This can be extended to triple integrals to find the mass of a solid, or the center of mass of it. But an extension to the area/volume perspective: triple integrals will give the 4D equivalent of volume., Here's answers... Consider the integral given. It runs on 0 to 8 for the outermost bound, then the next bound runs $\pm\sqrt{64-y^2}.$ That indicates the semicircle portion of the origin-centered circle of radius 8 that has positive y-coordinate., The latter expression is an iterated integral in spherical coordinates. Finally, in order to actually evaluate an iterated integral in spherical coordinates, we must of course determine the limits of integration in \(\phi\text{,}\) \(\theta\text{,}\) and \(\rho\text{.}\) The process is similar to our earlier work in the other two coordinate ...