Q 17. - Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Given Information

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

Step 2: 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 $\left[0,4\right]$

For type I region, $a=0, b=4$

Step 3: Computation of area

Evaluation of Area,

The area is evaluated as ${\iint }_{\Omega }dA={\int }_a^{b{g}_{g_1}\left(x\right)}{\int }_2^{g_2\left(x\right)}dydx$,

$\begin{array}{rcl}{\int }_0^4{\int }_{\frac{1}{2}x}^{\sqrt{x}}dydx& =& {\int }_0^4[y{]}_{\frac{1}{2}x}^{\sqrt{x}}dx\\ & =& {\left[\frac{2}{3}{x}^{3/2}-\frac{x^2}{4}\right]}_0^4\\ {\int }_0^4{\int }_{\frac{1}{2}x}^{\sqrt{x}}dydx& =& \frac{2}{3}\left(8\right)-4\\ & =& \frac{4}{3}\end{array}$

The required area $\frac{4}{3}$.