Question

Use polar coordinates to find the volume V of the solid region T.

T is inside the sphere

x² + y² + z² = 25

and inside the cylinder

x² + y² = 5y. V=?

Answer 1

To find the volume of the solid region T, we need to find the intersection between the sphere and the cylinder and then use polar coordinates to integrate over the region T. Evaluating the integral, the volume of T is 15π cubic units.

Step-by-step explanation:

To find the volume of the solid region T, we need to find the intersection between the sphere and the cylinder. Let's first find the equation of the intersection curve. Using polar coordinates, we substitute x = rcos(theta) and y = rsin(theta) into the equation of the cylinder: (rcos(theta))^2 + (rsin(theta))^2 = 5(rsin(theta)). Simplifying, we get r^2 = 5rsin(theta). This equation represents the intersection curve.

The limits of integration for r are 0 to 5sin(theta) and the limits of integration for theta are 0 to 2pi. So the volume of the solid region T can be found using the formula V = ∫∫∫ r dz dr dtheta, where the limits of integration are as mentioned before. Evaluating the integral, we get the volume of T = 15π cubic units.

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