Find the area enclosed by graphs of y=√x and y= x³?

Solution:

The two curves are represented on the diagram below:

The shaded area shown above is the area between the two curves OABCO. Mathematically the shaded area is given by 

= Area under the curve (y = √x)  minus area under the curve (y = x³)

= \(\int_{0}^{1}(\sqrt{x} - x^{3})dx\)

=\(\int_{0}^{1}\sqrt{x}dx - \int_{0}^{1}x^{3}dx\)

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

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

= 2/3  -  1/4 

=  (8  -  3) / 12

= 5/12

Find the area enclosed by graphs of y=√x and y= x³?

Summary:
The area enclosed by graphs of y=√x and y= x³  is 5/12.