Find the volume of the solid
formed by revolving the region
bound by the graphs ,
, and about the
-axis. Use the disk method.

Round your answer to the nearest thousandth.

Ask by Deleon Henry. in the United States

Mar 23,2025

Question

Find the volume of the solid
formed by revolving the region
bound by the graphs ,
, and about the
-axis. Use the disk method.

Round your answer to the nearest thousandth.

Ask by Deleon Henry. in the United States

Mar 23,2025

UpStudy AI · Accepted Answer

We first note that the region is bounded by

- $$ y = x^2 - 12 $$,
- $$ y = 0 $$,
- $$ x = -2 $$, and
- $$ x = 2 $$.

For any $$x$$ in $$[-2,2]$$, the $$y$$-coordinate of the curve is
$$
y = x^2-12,
$$
which is negative in this interval (since $$x^2 \leq 4$$ and thus $$x^2-12 \leq -8$$). The region between $$ y=x^2-12 $$ and $$ y=0 $$ is located below the $$x$$-axis. However, when we revolve the region about the $$x$$-axis, the vertical distance between $$y=0$$ and $$y=x^2-12$$ becomes the radius of a typical disk. That is, for a fixed $$x$$, the radius is
$$
R(x)=0-\bigl(x^2-12\bigr)=12-x^2.
$$

Using the disk method, the volume is given by
$$
V=\pi\int_{-2}^{2}\Bigl(R(x)\Bigr)^2\,dx=\pi\int_{-2}^{2}(12-x^2)^2\,dx.
$$

Since the integrand $$(12-x^2)^2$$ is an even function, we can simplify the integral by doubling the integral from 0 to 2:
$$
V=\pi\cdot2\int_{0}^{2}(12-x^2)^2\,dx.
$$

First, expand the integrand:
$$
(12-x^2)^2 = 144 - 24x^2 + x^4.
$$

Thus,
$$
V=2\pi\int_{0}^{2}\left(144 - 24x^2 + x^4\right)\,dx.
$$

Now, compute the antiderivative term by term:
$$
\int_{0}^{2}144\,dx = 144x\Big|_{0}^{2}=144(2)=288,
$$
$$
\int_{0}^{2}24x^2\,dx = 24\cdot\frac{x^3}{3}\Big|_{0}^{2} = 8x^3\Big|_{0}^{2}=8(8)=64,
$$
$$
\int_{0}^{2}x^4\,dx = \frac{x^5}{5}\Big|_{0}^{2} = \frac{32}{5}.
$$

Substitute these into the formula:
$$
\int_{0}^{2}\left(144-24x^2+x^4\right)dx = 288 - 64 + \frac{32}{5} = 224 + \frac{32}{5} = \frac{1120+32}{5}=\frac{1152}{5}.
$$

Thus, the volume is
$$
V=2\pi\cdot\frac{1152}{5}=\frac{2304\pi}{5}.
$$

To express the volume in a decimal approximation, we have
$$
V\approx \frac{2304\pi}{5}\approx 460.8\pi\approx460.8\times3.1416\approx1447.256.
$$

Rounded to the nearest thousandth, the volume is approximately
$$
V\approx 1447.256.
$$
The volume of the solid is approximately 1447.256 cubic units.

Find the volume of the solid
formed by revolving the region
bound by the graphs ,
, and about the
-axis. Use the disk method.

Round your answer to the nearest thousandth.

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Find the volume of the solid formed by revolving the region bound by the graphs , , and about the -axis. Use the disk method. Round your answer to the nearest thousandth.

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Find the volume of the solid
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, and about the
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Round your answer to the nearest thousandth.