Question

Let R be the region in the first quadrant bounded by the
graph of y = x² and the line y = 4.
(a) Find the area of the region R.
(b) A solid S has base R. Each cross-section of this solid
perpendicular to the y-axis is a semicircle, with its
diameter in R. Find the volume of the solid S.
(c) Another solid is formed by revolving R about the line
y = 7. Find the volume of this new solid.

Answer 1

a) The area within the region is $\frac{16}{3}$.

b) The volume of the solid $S$ is $\pi$ cubic units.

c) The volume of the solid of revolution is $\frac{164}{3}\pi$ cubic units.

How to apply integral formulas to determine areas and volumes

a) The region is bounded between $(0,0)$ and $(2, 4)$, the area of the region is defined by the following integral formula:

$A = \int\limits^{b}_{a} {[g(x)-f(x)]} \, dx$   (1)

Where:

• $f(x)$ - Lower function
• $g(x)$ - Upper function
• $a$ - Initial value
• $b$ - Final value

If we know that $f(x) = x^{2}$, $g(x) = 4$, $a = 0$ and $b = 2$, then the area within the region is:

$A = \int\limits^{2}_{0} {4-x^{2}} \, dx$

$A = \frac{16}{3}$

The area within the region is $\frac{16}{3}$. $\blacksquare$

b) The volume of the solid $S$ is determined by the following integral formula:

$V = \frac{\pi}{8} \int\limits^2_0 {x^{2}} \, dy = \frac{\pi}{8} \int\limits^2_0 {x^{2}\cdot (2\cdot x)} \, dx = \frac{\pi}{4}\int\limits^2_0 {x^{3}} \, dx$

$V = \pi$

The volume of the solid $S$ is $\pi$ cubic units. $\blacksquare$

c) The solid of revolution around an axis parallel to the $x$-axis is described by the following formula:

$V = \pi \int\limits^b_a {[[f(x)-k]^{2}-[g(x)-k]^{2}]} \, dx$   (2)

Where $k$ is the y-coordinate of the axis of revolution.

If we know that $f(x) = x^{2}$, $g(x) = 4$, $k = 7$, $a = 0$ and $b = 2$, then the volume of the solid of revolution is:

$V = \pi\int\limits^2_0 {[(x^{2}-7)^{2}-(4-7)^{2}]} \, dx$

$V = \frac{164}{3}\pi$

The volume of the solid of revolution is $\frac{164}{3}\pi$ cubic units. $\blacksquare$

To learn more on solids of revolution, we kindly invite to check this verified question: https://brainly.com/question/338504