Question

Find the volume V of the solid obtained by rotating the region bounded by the curves x = 2 - 5y, x = 0, y = 5 about the y-axis?

Answer 1

The volume V of the solid obtained by rotating the region bounded by the curves x = 2 - 5y, x = 0, y = 5 about the y-axis is
`\( V = (25)/(6)π \)` cubic units.

Step-by-step explanation:

To find the volume of the solid obtained by rotating the given region about the y-axis, we can use the disk method. The integral for the volume
`\( V \)` is given by the formula
`\( V = π ∫_(a)^(b) [f(y)]^2 \, dy \)` , where
`\( f(y) \)` represents the radius of the disk at each \( y \) value. In this case, the curves x = 2 - 5y, x = 0, and y = 5 bound the region. The limits of integration
`(\( a \) and \( b \))` are determined by the points of intersection of the curves.

The curves intersect at
`\( y = 1 \).` Therefore, the integral becomes
`\( V = π ∫_(0)^(1) [(2 - 5y)]^2 \, dy \)` . Simplifying this integral yields
`\( V = (25)/(6)π \)` cubic units. The volume
`\( V \)` represents the three-dimensional space enclosed by the rotated region.

Understanding and applying the disk method is crucial in calculus, especially in the context of finding volumes of solids of revolution. The method involves integrating cross-sectional areas to determine the overall volume of the rotated shape.