Question

# The integral represents the volume of a solid. Describe the solid. \pi\int_{0}^{1}(y^{4}-y^{8})dy a) The integral describes the volume of the solid ob

Integrals
The integral represents the volume of a solid. Describe the solid.
$$\pi\int_{0}^{1}(y^{4}-y^{8})dy$$
a) The integral describes the volume of the solid obtained by rotating the region $$R=\{\{x,\ y\}|0\leq y\leq1,\ y^{4}\leq x\leq y^{2}\}$$ of the xy-plane about the x-axis.
b) The integral describes the volume of the solid obtained by rotating the region $$R=\{\{x,\ y\}|0\leq y\leq1,\ y^{2}\leq x\leq y^{4}\}$$ of the xy-plane about the x-axis.
c) The integral describes the volume of the solid obtained by rotating the region $$R=\{\{x,\ y\}|0\leq y\leq1,\ y^{4}\leq x\leq y^{2}\}$$ of the xy-plane about the y-axis.
d) The integral describes the volume of the solid obtained by rotating the region $$R=\{\{x,\ y\}|0\leq y\leq1,\ y^{2}\leq x\leq y^{4}\}$$ of the xy-plane about the y-axis.
e) The integral describes the volume of the solid obtained by rotating the region $$R=\{\{x,\ y\}|0\leq y\leq1,\ y^{4}\leq x\leq y^{8}\}$$ of the xy-plane about the y-axis.

2021-05-18

Step 1
$$\pi\int_{0}^{1}(y^{4}-y^{8})\ dy=1$$
$$d1=\pi[(y^{2})^{2}-(y^{4})^{2}]\ dy$$
Axis of solution $$=y-axis$$
upper boundary $$x=y^{2}$$
lower boundary $$x=y^{4}$$
region in x-y plane
$$y^{4}<x<y^{2}$$
$$0\leq y\leq1$$