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Chapter 13: Q 17. (page 1014)

Use the results of Exercises 15 and 16 to find the area of the region $惟$ shown in Exercise 13.

Short Answer

Expert verified

Area of region is $\frac{4}{3}$ units.

Step by step solution

01

Given Information

The region is bounded by curves $y = \sqrt{x} and y = \frac{1}{2} x$

02

Consideration

The region may be considered at type I or type II.

When it is expressed as type I region, it is bounded by $y = \frac{1}{2} x$ below and $y = \sqrt{x}$ above for all $x$ in interval $[0, 4]$

For type I region, $a = 0, b = 4$

03

Computation of area

Evaluation of Area,

The area is evaluated as ${鈭�}_{惟} d A = {鈭�}_{a}^{b g_{g_{1}} (x)} {鈭�}_{2}^{g_{2} (x)} d y d x$ ,

$\begin{array}{rcl} {鈭�}_{0}^{4} {鈭�}_{\frac{1}{2} x}^{\sqrt{x}} d y d x & = & {鈭�}_{0}^{4} [y]_{\frac{1}{2} x}^{\sqrt{x}} d x \\ = & {[\frac{2}{3} x^{3 / 2} - \frac{x^{2}}{4}]}_{0}^{4} \\ {鈭�}_{0}^{4} {鈭�}_{\frac{1}{2} x}^{\sqrt{x}} d y d x & = & \frac{2}{3} (8) - 4 \\ = & \frac{4}{3} \end{array}$

The required area $\frac{4}{3}$ .

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