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
ANSWERED
asked 2021-05-17
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.

Answers (1)

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
\(y^{4}<x<y^{2}\)
\(0\leq y\leq1\)
Answer: (c)

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