### Home > CALC > Chapter 8 > Lesson 8.3.3 > Problem8-125

8-125.

Given: $f(x) =\sqrt { x - 4 }$

1. Find the area of the region bounded by $f$, $x$-axis, and the line $x = 8$.

Before you setup an integral, sketch the graph and shade the region. You should be able to sketch $f(x)$ without a graphing calculator.

2. Find the line $x = c$ that will divide the region from part (a) into two equal pieces.

Solve for $c$.

3. Find the volume of the solid that is formed by rotating the region described in part (a) about the $x$-axis.

Use disks.

4. Find a value $d$ such that a plane perpendicular to the $x$-axis at $d = x$ will divide the solid in part (c) into two equal parts.

Refer to the hint in part (b) and follow a similar process using a generic volume formula instead of a generic area formula.

$\text{Solve:}\int_4^d \pi(\sqrt{x-4})^2dx=\int_d^8\pi(\sqrt{x-4})^2dx$

$\frac{d^2}{2}-4d+8=-\frac{d^2}{2}+4d$