Question

Math

Posted 4 months ago

```
Let $R$ be the region enclosed by the line $y=2$ and the curve $2(x-1)^{2}$.
A solid is generated by rotating $R$ about the line $y=2$.
Which one of the definite integrals gives the volume of the solid?
Choose 1 answer:
(A) $\pi \int_{0}^{2}\left[2-2(x-1)^{2}\right] d x$
(B) $\pi \int_{0}^{2}\left[2(x-1)^{2}\right]^{2} d x$
(C) $\pi \int_{0}^{2}\left[2-2(x-1)^{2}\right]^{2} d x$
() $\pi \int_{0}^{2} 2(x-1)^{2} d x$
```

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Answer from Sia

Posted 4 months ago

Solution by Steps

step 2

The region $R$ is enclosed by the line $y=2$ and the curve $y=2(x-1)^2$

step 3

The volume of the solid of revolution can be found using the disk method. The formula for the volume is $V = \pi \int_{a}^{b} [R(x)]^2 dx$, where $R(x)$ is the distance from the axis of rotation to the curve

step 4

Here, $R(x) = 2 - 2(x-1)^2$

step 5

Therefore, the volume is given by $V = \pi \int_{0}^{2} [2 - 2(x-1)^2]^2 dx$

step 6

This matches option (C) in the multiple choice answers

C

Key Concept

Volume of a solid of revolution

Explanation

The volume of a solid generated by rotating a region about a line can be found using the disk method, which involves integrating the square of the distance from the axis of rotation to the curve.

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