Question:
Find the area under the given curves and given lines:
(i) $y=x^{2}, x=1, x=2$ and $x$-axis
(ii) $y=x^{4}, x=1, x=5$ and $x$-axis
Solution:
i. The required area is represented by the shaded area ADCBA as
Area $\mathrm{ADCBA}=\int_{1}^{2} y d x$
$=\int_{1}^{2} x^{2} d x$
$=\left[\frac{x^{3}}{3}\right]_{1}^{2}$
$=\frac{8}{3}-\frac{1}{3}$
$=\frac{7}{3}$ units
ii. The required area is represented by the shaded area ADCBA as\
Area $\mathrm{ADCBA}=\int_{1}^{5} x^{4} d x$
$=\left[\frac{x^{5}}{5}\right]_{1}^{5}$
$=\frac{(5)^{5}}{5}-\frac{1}{5}$
$=(5)^{4}-\frac{1}{5}$
$=625-\frac{1}{5}$
$=624.8$ units