Q 13. - Calculus

Question

Following region is bounded by functions $y=\frac{1}{2}x\text{ and }y=\sqrt{x}$.

Express $\Omega$ as type I or type II region. If $\Omega$ is a type I region, what are

$a, b$? If $\Omega$ is a type II region, what are $c, d$?

Step-by-Step Solution

Answer

For type I region, $a=0, b=4$

For type II region, $c=0, d=2$

Step 1: Given Information

The given curves are $y=\sqrt{x}\text{ and }y=\frac{1}{2}x$

Step 2: Simplification

It may be considered as either type I or type II.

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

Hence, for type I region, $a=0, b=4$

When region is expressed and type II, it is bounded by $y=\sqrt{x}$ to left and $y=\frac{1}{2}x$ for all $y$ in interval $\left[0,2\right]$.

Hence, for type II region $c=0, d=2$