Question

1)Find the area under the given curves and given lines:
$y=x^2,x=1,x=2\text{and}x-axis$

2)Find the area under the given curves and given lines:
$y=x^4,x=1,x=5\text{and}x-axis$

Answer 1

1) Given: $y=x^2,x=1,x=2\text{and}x-axis$
Draw figure

Required area = Area ABCDA
$={\int }_1^{2}ydx$

$={\int }_1^2{x}^{2}dx$

$={\left[\frac{x^3}{3}\right]}_1^2=\frac{1}{3}[2^3-1^3]$

$=\frac{1}{3}[8-1]=\frac{7}{3}$

Final answer:
Therefore, required Area = $\frac{7}{3}$ square units.

2)Given: $y=x^4,x=1,x=5\text{and}x-axis$

Area required = Area ABCDA

$={\int }_1^{5}ydx$

$={\int }_1^5{x}^{4}dx$

$={\left[\frac{x^5}{5}\right]}_1^5=\frac{1}{5}[5^5-1^5]$

$=5^4-\frac{1}{5}$

$=625-0.2=624.80$

Final answer:
Therefore, required Area =$624.8$ square units.