and above by the sphere

babeeb0oL
2021-03-05
Answered

Consider the solid that is bounded below by the cone $z=\sqrt{3{x}^{2}+3{y}^{2}}$

and above by the sphere${x}^{2}+{y}^{2}+{z}^{2}=16.$ .Set up only the appropriate triple integrals in cylindrical and spherical coordinates needed to find the volume of the solid.

and above by the sphere

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averes8

Answered 2021-03-06
Author has **92** answers

Step 1

To set the triple integral in cylindrical coordinates

By using relation,

Thus,

The cone

And the sphere become

To find the limit of r,

Consider,

Step 2

Thus, we can describe the region as

Hence, the triple integral for the volume by cylindrical coordinates is

Step 3

Now, to set the triple integral in spherical coordinates

Since,

The sphere

From the cone

Step 4

Thus, we can describe the region as

Hence, the triple integral for the volume of the solid by spherical coordinate is

Step 5

Now, evaluating the integral of cylindrical coordinate we get.

And evaluating the integral of spherical coordinate

Thus, by both coordinate systems, we get the same volume.

Therefore, both triple integrals are appropriate.

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