Question

Find the volume of the solid obtained by rotating the region in the first quadrant bounded by y=x 7 .y=1, and the y axis about the line y=−3 Volume =

Answer 1

The volume of the solid obtained by rotating the region in the first quadrant bounded by
`\(y = x^7\)`, y = 1, and the y-axis about the line y = -3 is
`\((442)/(11)\)` cubic units.

Step-by-step explanation:

To find the volume of the solid of revolution, we use the disk method. The region in the first quadrant is bounded by
`\(y = x^7\), \(y = 1\)`, and the y-axis. We rotate this region about the line y = -3. The radius of the disks is the distance between the line y = -3 and the curves
`\(y = x^7\)` and y = 1.

The limits of integration are determined by the intersection points of
`\(y = x^7\)` and y = 1, which are x = 1. The radius r is the difference between the y-coordinate of the axis of rotation and the y-coordinate of the curve, sor = 1 - (-3) = 4. The volume V is given by the integral:

`\[V = \pi \int_(0)^(1) (4 - x^7)^2 \,dx.\]`

Evaluating this integral yields the final answer of
`\((442)/(11)\)`cubic units.