The region between the curve $$ y=\sqrt{4-x^{2}} $$ and the $$ x $$-axis is revolved about the $$ x $$-axis to generate a solid. Find the exact value of its volume. $$ \text { [?] } $$

Question

UpStudy AI · Accepted Answer

We start with the curve
$$
y=\sqrt{4-x^2},
$$
which represents the upper half of a circle of radius 2. The region between this curve and the $$ x $$-axis (i.e. $$ y\ge0 $$) is bounded, for $$ x $$, by
$$
-2\le x\le2.
$$

When this region is revolved about the $$ x $$-axis, we can use the disk method to find the volume. Each slice perpendicular to the $$ x $$-axis is a disk with radius
$$
R(x)=\sqrt{4-x^2},
$$
so its area is
$$
A(x)=\pi\left(R(x)\right)^2=\pi\left(4-x^2\right).
$$

The volume $$ V $$ is given by
$$
V=\int_{-2}^{2}A(x)\,dx=\pi\int_{-2}^{2}(4-x^2)\,dx.
$$

We now compute the integral:
$$
\int_{-2}^{2}(4-x^2)\,dx.
$$

First, find an antiderivative:
$$
\int(4-x^2)\,dx=4x-\frac{x^3}{3}+C.
$$

Evaluating the definite integral from $$-2$$ to $$2$$:
$$
\left[4x-\frac{x^3}{3}\right]_{-2}^{2}=\left(4(2)-\frac{2^3}{3}\right)-\left(4(-2)-\frac{(-2)^3}{3}\right).
$$

Calculate each part:
$$
\text{At } x=2:\quad 4(2)-\frac{8}{3}=8-\frac{8}{3}=\frac{24-8}{3}=\frac{16}{3},
$$
$$
\text{At } x=-2:\quad 4(-2)-\frac{-8}{3}=-8+\frac{8}{3}=\frac{-24+8}{3}=-\frac{16}{3}.
$$

Thus,
$$
\left[4x-\frac{x^3}{3}\right]_{-2}^{2}=\frac{16}{3}-\left(-\frac{16}{3}\right)=\frac{32}{3}.
$$

Substitute back into the volume:
$$
V=\pi\cdot\frac{32}{3}=\frac{32\pi}{3}.
$$

Therefore, the exact value of the volume is
$$
\boxed{\frac{32\pi}{3}}.
$$
The volume of the solid generated by revolving the region between $$ y = \sqrt{4 - x^2} $$ and the $$ x $$-axis about the $$ x $$-axis is $$ \frac{32\pi}{3} $$.

The region between the curve

and the -axis is revolved about
the -axis to generate a solid.
Find the exact value of its volume.

Upstudy AI Solution

Answer

Solution

Beyond the Answer

Related Questions

Latest Calculus Questions

The region between the curve and the -axis is revolved about the -axis to generate a solid. Find the exact value of its volume.