Let r be the region bounded by the graphs of y= √x  and y=x/2, how do you find the area of R?

Solution:

The area bounded by the curves is shown below:

The area of the bounded curve (shaded area) is given by the following expression:

Area bounded by the two curves 

= \(\int_{0}^{4}(\sqrt{x} - \frac{x}{2})dx\)

= \(\int_{0}^{4}\sqrt{x}dx - \int_{0}^{4}\frac{x}{2}dx\)

=\(\int_{0}^{4}x^{\frac{1}{2}}dx - \int_{0}^{4}\frac{x}{2}dx\)

= \([\frac{x}{\frac{3}{2}}]_{0}^{4} - \frac{1}{2}[\frac{x^{2}}{2}]_{0}^{4}\)

= \(\frac{2}{3}[(4^{\frac{3}{2}} - (0)^{\frac{3}{2}})] - \frac{1}{4}[4^{2} - 0^{2}]\)

= \(\frac{2}{3}[8] - \frac{1}{4}[16]\)

= 16/3 - 4

= (16 -12)/3

= 4/3 units²

4/3 units² is the area bounded by the curves.

Let r be the region bounded by the graphs of y= √x  and y=x/2, how do you find the area of R?

Summary:
The regionbounded by the graphs of y= √x  and y=x/2 has the area of 4/3 units²