# Find the volume of the solid bounded above by the sphere x^2 + y^2 + z^2 = 64 and below by the...

## Question:

Find the volume of the solid bounded above by the sphere {eq}x^2 + y^2 + z^2 = 64 {/eq} and below by the cone {eq}z = \sqrt{x^2 + y^2} {/eq}.

## Cylindrical Coordinates:

The problem above is best handled using some form of polar coordinates: either cylindrical or spherical. Because they are typically easier to work with, we will be using cylindrical coordinates. Recall

{eq}x = r \cos \theta {/eq}

{eq}y = r \sin \theta {/eq}

{eq}z = z {/eq}

{eq}r^2 = x^2+y^2 {/eq}

{eq}\theta = \tan^{-1} \frac{y}{x} {/eq}

{eq}dV = r \ dz \ dr \ d\theta {/eq}

## Answer and Explanation: 1

Become a Study.com member to unlock this answer! Create your account

View this answerIn cylindrical coordinates we have {eq}r \leq z \leq \sqrt{8-r^2} {/eq}. Next, {eq}r {/eq} is bounded by the intersection of the surfaces:

{eq}\beg...

See full answer below.

#### Ask a question

Our experts can answer your tough homework and study questions.

Ask a question Ask a question#### Search Answers

#### Learn more about this topic:

from

Chapter 13 / Lesson 10Learn how to convert between Cartesian, cylindrical and spherical coordinates. Discover the utility of representing points in cylindrical and spherical coordinates.

#### Related to this Question

- Find the volume of the solid bounded below by the half-cone z= sqrt(x^2 + y^2) and above by the sphere x^2 + y^2 + z^2 = 16.
- Compute the volume of the solid bounded by the sphere x^2 + y^2 + z^2 = 4 and the cone z = sqrt(x^2 + y^2).
- Compute the volume of a solid bounded below by the sphere x^2 + y^2 + z^2= a^2 and above by the cone z = sqrt(x^2 + y^2).
- Compute the volume of the solid bounded above by the sphere x^2 + y^2 + z^2 = 100 and below by the cone z = sqrt(x^2 + y^2).
- Compute the volume of the solid bounded above by the sphere x^2 + y^2 + z^2 = 1 and below by the cone z = sqrt(x^2 + y^2).
- Find the volume of the solid bounded below by the cone z = \sqrt{x^2 + y^2} and above by the sphere x^2 + y^2 + z^2 = 4
- Find the volume of the solid that is enclosed by the cone z = sqrt(x^2 + y^2) and the sphere x^2 + y^2 + z^2 = 128.
- Find the volume of the solid which is above the cone z = sqrt(x^2 + y^2) and inside the sphere given by x^2 + y^2 + z^2 = 18.
- Find the volume of the solid region bounded below by the cone \sqrt{x^2+y^2} = z and above by the sphere x^2+y^2+z^2=4^2
- Find the volume of the solid bounded below by the circular cone z = 1.5 square root x^2 + y^2 and above by the sphere x^2 + y^2 + z^2 = 2.5 z.
- Compute the volume of the solid which is above the cone z = sqrt(x^2 + y^2) and inside the sphere given by x^2 + y^2 + z^2 = 50.
- Compute the volume of the solid which is above the cone z = sqrt(x^2 + y^2) and inside the sphere given by x^2 + y^2 + z^2 = 18.
- Compute the volume of the solid bounded below by the half-cone z = sqrt(x^2 + y^2) and above by the sphere x^2 + y^2 + z^2 = 16.
- Find the volume of the solid D bounded below by the cone z = sqrt(x^2 + y^2) and above by the sphere rho = cos(phi).
- Find the volume of the solid region bounded by the cone z=\sqrt{(3x^2 + 3y^2)} and the sphere x^2+y^2+z^2=9
- Find the volume of the solid that is enclosed by the cone z = \sqrt{x^2 + y^2} and the sphere x^2 + y^2 + z^2 = 2.
- Find the volume of the solid that is enclosed by the cone z = sqrt{x^2+y^2} and the sphere x^2 + y^2 + z^2 = 162.
- Find the volume of the solid that is enclosed by the cone z = \sqrt{x^{2} + y^{2 and the sphere x^{2} + y^{2} + z^{2} = 72
- Calculate the volume of the solid bounded by the sphere x^2+y^2 + z^2 = 3^2 and the cone z = \sqrt{x^2+y^2}
- Find the volume of the bounded region above by the sphere x^2+y^2+z^2=64 and below by the cone z=\sqrt{x^2+y^2}.
- Find the volume of the solid region which is bounded below by the cone z = \sqrt{x^2+y^2} and lies within the sphere x^2 + y^2 + z^2 = 16
- Find the volume of the solid that lies within the sphere x^{2} + y^{2} + z^{2} = 1, above the xy-plane, and outside the cone z = 8 \sqrt{x^{2} + y^{2.
- Find the volume of the solid that lies within the sphere x^2 + y^2 + z^2 = 81, above the xy-plane, and outside the cone z = 6 sqrt(x^2 + y^2).
- Find the volume of the solid that lies within the sphere x^2 + y^2 + z^2 = 4, above the xy-plane, and below the cone z = sqrt(x^2 + y^2).
- Find the volume of the solid that lies within the sphere x^2 + y^2 + z^2 = 49, above the xy-plane, and below the cone z = sqrt(x^2 + y^2).
- Using calculus, compute the volume of the solid above the cone z = sqrt(x^2 + y^2) and below the sphere x^2 + y^2 + z^2 = 1.
- Compute the volume of the solid that lies above the cone z = sqrt(x^2 + y^2) and below the sphere x^2 + y^2 + z^2 = z.
- Find the volume of the portion of the sphere = 3 bounded below by the cone = 3 ? .
- Calculate the volume of the solid which is above the cone z = sqrt(x^2 + y^2) and inside the sphere given by x^2 + y^2 + z^2 = 18.
- Find the volume of the solid bounded above by sphere \rho =1 and below by cone z= \frac{r}{\sqrt 3}
- Find the volume of the solid bounded below by the upper nappe of the cone z^2 = x^2 + y^2 and bounded above the sphere x^2 + y^2 + z^2 = 9.
- Find the volume of the solid that is enclosed by the cone z = square root {x^2 + y^2} and the sphere x^2 + y^2 + z^2 = 32.
- Find the volume of the region bounded above by the sphere x^2 + y^2 + z^2 = 49 and below the cone z = \sqrt{x^2 + y^2}.
- Find the volume of the region bounded above by the sphere x^2 + y^2 + z^2 = 9 and below by the cone z= \sqrt{x^2 + y^2}. A) 9 \pi (2-\sqrt{2}). B) 9 \pi (2-\sqrt{3}). C) \frac{27}{4} \pi (2 - \sqrt{3
- Find the volume of the solid bounded below by the sphere rho = 2cos(phi) and above by the cone z = sqrt(3x^2 + 3y^2).
- Find the volume of the region enclosed by the cone z = sqrt(x^2 + y^2) and the sphere x^2 + y^2 + z^2 = 4.
- Find the volume of the solid bounded above by the sphere x^2 + y^2 + z^2 = 9 and below by the half-cone z = sqrt(2x^2 + 2y^2).
- Use polar coordinates to find the volume of the solid bounded above by the sphere x^2 + y^2 + z^2 = 2 and below by the cone z = sqrt(x^2 + y^2).
- Use polar coordinates to find volume of the solid bounded above by the sphere x^2 + y^2 + z^2 = 2 and below by the cone z = sqrt(x^2 + y^2).
- Find the volume of the solid bounded above and below by the cone z^2 = x^2 + y^2, and the side by y = 0 and y = sqrt(4 - x^2 - z^2).
- Find the volume between the cone z = sqrt(x^2 + y^2) and the sphere x^2 + y^2 + z^2 = 4.
- Use polar coordinates to find the volume of the solid bounded above by the cone z = sqrt(x^2 + y^2) and below by the sphere x^2 + y^2 + z^2 = 1.
- Find the volume of the solid which is above the cone \displaystyle z=\sqrt{x^2+y^2} and inside the sphere given by \displaystyle x^2+y^2+z^2=50.
- Find the volume of the solid which is above the cone z = Squareroot x^2 + y^2 and inside the sphere given by x^2 + y^2 + z^2 = 98.
- Find the volume of the solid outside the cylinder x^2 + y^2 = 1 that is bounded above by the sphere x^2 + y^2 + z^2 = 8 and below by the cone z = square root (x^2 + y^2).
- Find the volume of the solid that lies above the cone z = \sqrt {x^2 + y^2} and below the sphere x^2 + y^2 + z^2 = z.
- Find the volume of the solid that lies above the cone z= \sqrt{x^2+y^2} and below the sphere.
- Find the volume of the solid that lies above the cone z = \sqrt{x^2+y^2} and below the sphere x^2+y^2+z^2=6z
- Find the volume of the solid that lies above the cone z= \sqrt{(x^2+y^2)} and below the sphere x^2+y^2+z^2=r^2
- Compute the volume of the solid that lies within the sphere x^2 + y^2 + z^2 = 4, above the xy-plane, and below the cone z = sqrt(x^2 + y^2).
- Find the volume of the solid bounded below by the circular cone z = 1.5*sqrt(x^2 + y^2) and above by the sphere x^2 + y^2 + z^2 = 2.75.
- Calculate the volume of the solid that lies within the sphere x^2 + y^2 + z^2 = 4, above the xy-plane, and below the cone z = sqrt(x^2 + y^2).
- Find the volume of the solid bounded by the sphere x^2 + y^2 + z^2 = 4, and the paraboloid z = x^2 + y^2.
- Use cylindrical coordinates to find exactly the volume of the solid that is bounded above by the sphere x^2 + y^2 + z^2 =1 and below by the cone z= \sqrt{x^2+y^2}
- A solid is bounded above by a sphere x^2 + y^2 + z^2 = 1 and below by the cone z = (x^2 + y^2)^{\frac{1}{2 . Use cylindrical coordinates to find the exact volume of the solid.
- Find the volume of the solid that lies within the sphere x^2 + y^2 + z^2 = 81, above the xy-plane, and outside the cone z = 8 square root{x^2 + y^2}.
- Find the volume of the solid that lies above the cone z = square root (x^2 + y^2) and below the sphere x^2 + y^2 + z^2 = z.
- Find the volume of the solid which is above the cone z = square root (x^2 + y^2) and inside sphere given by x^2 + y^2 + z^2 = 50.
- Find the volume of the solid that lies within the sphere x^2+y^2+z^2 = 144 , above the xy-plane and below the following cone. z = \sqrt{3x^2+3y^2}
- Find the volume of the solid that lies within the sphere x^2 + y^2 + z^2 = 81, above the xy-plane, and outside the cone z = 7sqrt(x^2 + y^2).
- Find the volume of the solid that lies within the sphere x^2+y^2+z^2=36, above the xy plane, and outside the cone z=\sqrt[5]{x^2+y^2}.
- Find the volume of the solid that lies within the sphere x^2+y^2+z^2=64 above the xy- plane, and below the cone z= \sqrt{x^2+y^2} .
- Find the volume of the solid that lies within the sphere x^2 + y^2 + z^2 = 36 , above the xy plane, and outside the cone z = \frac{6}{\sqrt{x^2+y^2 .
- Find the volume of the solid that lies within the sphere x^2+y^2+z^2=81, above the xy plane, and outside the cone z=7x^2+y^2.
- Find the volume of the solid that lies within the sphere x ^2 + y ^2 + z ^2 = 9 , above the x y plane, and outside the cone z = 5 ? x ^2 + y^ 2
- Find the volume of the solid that lies within The sphere x^2+y^2+z^2=16, above the xy plane, and outside the cone z=4(\sqrt{(x^2+y^2)}).
- FInd the volume of the solid that lies within the sphere x^2 + y^2 + z^2 =64 , above the xy-plane, and outside the cone z = 4 \sqrt{x^2 + y^2}
- Find the volume of the solid that lies within the sphere x2 + y2 + z2 = 36, above the xy-plane, and below the cone z=sqrt(x^2+y^2)
- Find the volume of the solid that lies within the sphere x^2 + y^2 + z^2 = 81, above the xy-plane, and below the cone z = x^2 + y^2.
- Find the volume of the solid that lies within the sphere x^2+y^2+z^2=81, above the xy plane, and outside the cone z=3 \sqrt{x^2+y^2}.
- Find the volume of the solid that lies within the sphere x^2+y^2+z^2=49 , above the xy-plane, and outside the cone z=6\sqrt{x^2+y^2} .
- FInd the volume of the solid that lies within the sphere x^2+y^2+z^2=25 above the xy-plane, and outside the cone z=6 \sqrt{x^2+y^2}
- Find the volume of the solid that lies within the sphere x^2 +y^2 +z^2 =49 , above the xy-plane, and outside the cone z=3\sqrt{x^2 +y^2} .
- Find the volume of the solid that lies within the sphere x^2 + y^2 + z^2 = 16 , above the xy plane, and outside the cone z = 7 \sqrt {x^2 + y^2}.
- Find the volume of the solid that lies within the sphere x^2 + y^2 + z^2 = 81, above the xy-plane, and below the cone z = y^2 + x^2.
- FInd the volume of the solid that lies within the sphere x^2+y^2+z^2=49 above the xy plane, and lies outside the cone z=2 \sqrt{x^2+y^2}
- Find the volume of the solid that lies within the sphere x^2+y^2+z^2 = 49 , above the xy plane, and outside the cone z = \sqrt{x^2+y^2}
- Find the volume of the solid that lies within the sphere x^2+y^2+z^2=49 , above the xy-plane, and outside the cone z=4\sqrt{x^2+y^2}
- 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 following cone. z = sqrt(x^2 + y^2)
- Find the volume of the solid that lies within the sphere x^2+y^2+z^2 = 36 , above the xy -plane, and outside the cone z=6 \sqrt{x^2+y^2} .
- Find the volume of the solid that lies within the sphere x^2 + y^2 + z^2 = 1|,a bove the xy-plane, and outside the cone z = 6 \sqrt {x^2 + y^2}|.
- Find the volume of the solid that lies within the sphere x^2 + y^2 + z^2 = 16|, above the xy| plane, and outside the cone z = 8 \sqrt {x^2 +y^2}|.
- FInd the volume of the solid that lies within the sphere x^2+y^2+z^2=49 above the xy plane, and outside the cone z=2\sqrt{x^2+y^2}
- Find the volume of the solid that lies within the sphere x^2+y^2+z^2=36, above the xy-plane, and below the cone z=\sqrt{x^2+y^2}.
- Find the volume of the solid that lies within the sphere x^2+y^2+z^2=9 , above the xy- plane, and outside the cone z=\sqrt{x^2+y^2} .
- Find the volume of the solid that lies within the sphere x^2+y^2+x^2=64, above the xy-plane, and outside the cone z=6\sqrt{x^2+y^2}.
- Find the volume of the solid that lies within the sphere x^2+y^2+z^2=49 above the xy plane, and outside the cone z=7 \sqrt{x^2+y^2}
- Find the volume of the solid that lies within the sphere x^2 + y^2 + z^2 = 144, above the xy-plane, and below the following cone. z = ? {(1/3) x^2 + (1/3) y^2} , V = ?
- Find the volume of the solid that lies within the sphere x^2 + y^2 + z^2 = 9, above the xz-plane, and below the cone y = sqrt(x^2 + z^2).
- Find the volume of the solid that lies within the sphere x^2 + y^2 + z^2 = 144 , above the xy - plane, and below the following cone z = \sqrt{3x^2+3y^2} .
- Find the volume of the solid that lies within the sphere x^{2} + y^{2} + z^{2} = 9, above the xz-plane, and below the cone y=\sqrt{x^{2}+z^{2.
- Find the volume of the solid that lies within the sphere x^2 + y^2 + z^2 = 1, above the xy plane, and outside the cone z = 4\sqrt{(x^2 + y^2)}.
- Using calculus, find the volume of the solid above the cone z = \sqrt{x^2+y^2} and below the sphere \rho =2
- Find the volume bounded by the intersection of the sphere x^2+y^2 + z^2 = 2z and the cone z^2 = x^2 + y^2
- Using calculus, find the volume of the solid above the cone z=\sqrt{x^2+y^2} and below the sphere x^2+y^2+z^2=1.
- Find the volume of the solid that lies within the sphere x^2+y^2+z^2=25 , above the xy plane, and outside the cone z=6 \sqrt {(x^2+y^2)}.
- Find the volume of the solid that lies within the sphere x2 + y2 + z2 = 4, above the xy plane, and outside the cone z = 2 sqrt x2 + y2.
- Find the volume of the solid that lies within the sphere x^2 + y^2 + z^2 = 64, above the xy-plane, and outside the cone z = 6*sqrt(x^2 + y^2).
- Find the volume of the solid that lies within the sphere x^2 + y^2 + z^2 = 4, above the xy-plane, and outside the cone z = 6*sqrt(x^2 + y^2).
- Find the volume of the solid that lies within the sphere x^2 + y^2 + z^2 = 16, above the xy-plane, and outside the cone z = 6*sqrt(x^2 + y^2).