Answered: 8. Using spherical coordinates,…

8. Using spherical coordinates, evaluate the triple integral x² + y² + z² dV D where D is the portion of the solid sphere x² + y² + z² < 1 that lies in the first octant.

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

Chapter10: Analytic Geometry

Section10.6: The Three-dimensional Coordinate System

Problem 41E: Does the sphere x2+y2+z2=100 have symmetry with respect to the a x-axis? b xy-plane?

See similar textbooks

Related questions

Q: SSL. xy dV, where T is the first-octant solid bounded by the coordinate planes and the upper half of…

A: Here we have to find ∫ ∫ ∫T xy dV , where T is the first-octant solid bounded by the coordinate…

Q: 1. Let S be the part of the sphere x2 + y? + z² plane z = 3, oriented so that the "top" of the…

A: Given the equation of sphere that lies above the plane z=3 x2+y2+z2=25

Q: Using spherical coordinates, the volume of the solid inside the sphere x+y+z -9 and above the plane…

Q: Evaluate fff 1- x2 - y2-z2 dx dy dz where V is the volume of the sphere x? + y? + z2 = a? by…

Q: 5. Use polar coordinates to find the volume of the solid enclosed by the hyper- boloid –r² – y² + z²…

A: To obtain the limits of integration for z, rewrite the equation -x2-y2+z2=1, and find z.…

Q: Consider the solid region E inside the sphere x² + y² + z² = 4 and outside the cylinder x² + y² = 1.…

A: This question is based on 3-D geometry.

Q: Let S be the part of the hyperbolic paraboloid z = x2- y2 located between the cylinders x2 + y? = 1…

Q: FIs () Use spherical coordinates to evaluate 1- av where the solid E lies above the cone z=V+y*,…

A: We have to evaluate.I=∭Exzdv, using spherical coordinate Consider, where the solid E lies above…

Q: 5. Evaluate the surface integral S_(x+ y) ds ; o is the part of the plane 2x + y +z = 3 that lies in…

Q: 2. The solid outside the cylinder x² + y² = 1 that is bounded above by the sphere x² + y? + x? = 8…

A: The solution is given below

Q: 2. Use a parametrization to express the area of the given surface as a double integral and then…

Q: Evaluate the surface integral. + y2 + z2) ds S is the part of the cylinder x² + y2 = 16 that lies…

A: We will solve this problem in some steps

Q: Find the volume of the solid below

A: Given that the solid bounded below by the xy-plane, on the side by the sphere having radius 2, and…

Q: Evaluate the surface integral. y ds S is the part of the paraboloid y = x² + z² that lies inside the…

A: The projection of S onto the xz-plane is the disk D=x,z: x2+z2≤9. We have to calculate the surface…

Q: 9. Find the area of the part of the paraboloid z = r² + y² that lies under the plane z = (Hint: Use…

A: Consider the provided question, My objective is to find the area of the part of the paraboloid…

Q: (9) Evaluate the surface integral / S, czdS, where o is the part of the plane r+2y+3z = 6 that lies…

Q: Let E be the solid region in the first octant bounded by the coordinate planes, the cylinder…

A: The cylinder x2+y2=5y and the sphere x2+y2+z2=25. Obtain the triple integral: ∫∫∫Ey dV The solid is

Q: Let D be the region enclosed by the surfaces z =-/4-x² - y and x² +y* . Determine JI| zdv by using…

Q: 1. Find the volume of the solid bounded by the paraboloids z = 6- x² - y and z = 2x? + 2y2.Use polar…

Q: Evaluate the surface integral. y² ds, S is the part of the sphere x² + y² + z² = 1 that lies above…

Q: dv Use spherical Coordinares to compute H 2 where W is the solid bounded by the spheres 4 ty+z² =g…

A: Given I=∫∫∫wdVx2+y2+z2Where W is the solid bounded by the spheres x2+y2+z2=9 and x2+y2+z2=4.

Q: (b) Verify Stokes' theorem on a region which is a portion of the sphere x? +y? +z² = 16 which lies…

Q: (b) Verify Stokes' theorem on a region which is a portion of the sphere r2 +y² +2 16 which lies…

Q: 17. Let T be the solid bounded below by the half-cone z = Vx² + y? and above by the spherical…

Q: 4. Find the volume of the solid bounded by the plane z = 0 and the paraboloid z=1-x² - y². (if using…

A: Solution :-

Q: (a) Let D be the region enclosed by the surfaces z =- |4–x² – y² and z= +y² . Determine zdv by using…

A: Triple integral is used to integrate the function of three variable over the three-dimensional…

Q: 2. Use polar coordinates to find the volume of the solid under the paraboloid z = x² + y² and above…

A: Let's find volume of solid by using polar coordinates.

Q: #3. Evaluate the surface integral xz ds, where S is the part of the plane 2x +2y +z = 4 that lies in…

Q: 2. Use Stokes' Theorem to evaluate , Vx F ·n dS where F(x, y, z) = and C is the part of the sphere…

Q: Let G be the solid in the first octant bounded by the sphere 2x? + 2y? + 2z? = 32 and the coordinate…

Q: 5. Use spherical coordinates to evaluate the triple integral ///+y²+²dV, where E is the ball: x? +y?…

Q: 1. Consider the surface which is the part of the sphere r? + y? + 2? = 4 that lies above the cone z…

Q: Consider the solid D that lies inside the cone z = and bounded by the spheres 3 y رد جر+ e][] p = 2…

Q: 5. Use the spherical coordinates to evaluate the integral f dzdydr, where R is the region enclosed…

A: In spherical coordinates, x = ρsinφsinθy= ρsinφcosθz= ρcosφ

Q: 1. Find the surface area of the cone z = Jx2 + y2 that lies between the plane y = x and the cylinder…

A: Surface area in xy-plane

Q: b) Write the equation of the sphere x² + y² + (z – 1)² = 1 in spherical coordinates then find the…

Q: 4. Evaluate the integral z dV, where D is the region bounded by the sphere x2 + y² + (z – 1)² = 1.…

A: Given: To evaluate the given integral as follows,

Q: 9. Using Stoke's Theorem Evaluate [(2x – y)dx – yz dy – y zdz) where c is the circle x +y =1,…

Q: Use spherical coordinates. Evaluate z dv, where E lies between the spheres x2 + y2 + z2 = 9 and x2 +…

A: Given equations of spheres: x2+y2+z2=9 x2+y2+z2=36 For spherical coordinates: x=ρ cosθ sinϕy=p sinθ…

Q: 2. Use a parametrization to express the area of the given surface as a double integral and then…

A: Given ,x2+y2+z2 = 22 _________(I)and z = x2+y2 __________(II)Putting the value of z from…

Q: 16. classify the surface: * 1- y? = 5x2 + 6z? Sphere Elliptic paraboloid Cone Plane Ellipsoid

Q: Let D be the solid region that lies above the cone z = x² + y² and below the sphere x+y²+= = =. The…

Q: (b) the volume of the solid B that lies above the cone z = 3r² + 3y² and below the sphere x² + y² +…

Q: 1. Find the surface area of the part of the sphere x+ y +z = 4 that lies above the plane z =1.

Q: A solid lies above the cone z = V x² + y2 and below the sphere x2 + y2 + z2 = z. Write a description…

A: We have, z=x2+y2 ⋯⋯1 Let us consider, x=ρsinϕcosθy=ρsinϕsinθz=ρcosϕ On substituting in…

Q: 17. Let T be the solid bounded below by the half-cone z = Vr2 + y? and above by the spherical…

A: Consider T be the solid bounded below by the half cone z=x2+y2 and above by the sphere x2+y2+z2=1.…

Q: 10. Use polar coordinates to find the volume of the solid under the surface z = (x² + y²)² and above…

A: The polar coordinate conversion is as follows x = r cos(θ) y = r sin(θ) x2 + y2 = r2 Under the…

Q: Evaluate the surface integral. v² as S is the part of the sphere x2 + y2 + z2 = 64 that lies inside…

A: A scalar function f(x,y,z) is a rule that is used to associate a number or quantity with a…

Q: 10. Use cylindrical coordinates to evaluate SS S(x+y+z) dV where E is the solid enclosed by the…

A: The region

Question

8. Using spherical coordinates, evaluate the triple integral
x² + y² + z² dV
D
where D is the portion of the solid sphere x² + y² + z² < 1 that lies in the first octant.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step by step

Solved in 2 steps

SEE SOLUTION Check out a sample Q&A here

Similar questions

Does a right circular cylinder such as an aluminum can have a symmetry with respect to at least one plane? b symmetry with respect to at least one line? c symmetry with respect to a point?
In isosceles trapezoid WXYZ with bases ZY and WX, ZY = 8, YX = 10 and WX = 20. Find height h the length of ZD or YE. Exercises 19, 20
Consider two spheres, defined by x^2 + y^2 + z^2 = 1 and (x − 2)^2 + (y − 6)^2 + (z − 3)^2 = 16, respectively. How close are the spheres from touching?
Consider a sphere E with center at the origin and radius equal to 1, following the Beltrami-Klein model, solve: to When X2+Y2<=1, find the point F(x,y) that is the intersection lxy with the sphere E
Let S be the solid inside the sphere x2+y2+(z-3)2 = 9 and the below the cone x2+y2 = z2. Setup the volume of S using spherical coordinates
Evaluate the surface integral. ∫∫sxdS, S is the part of the plane 6x+3y+z=6 that lies in the first octant.
Express the area of the given surface as an iterated double inegral in polar coordiantes, and then find the surface area: The portion of the sphere x^2+y^2+z^2=16 between the planes z=1 and z=2
Which is the equation of a sphere with a center at (1,0,-2) with a volume of 36(pi)(u^3)? a. x^2+y^2+z^2-2x+4z = 4 b. x^2+y^2+z^2+2x-4z = 4 c. x^2+y^2+z^2-2x+4z = -2 d. x^2+y^2+z^2+2x-4z = 2