Question

Using integration, find the area of the region bounded by the curves $y^2=4x,x=1,x=4$.

Answer 1

$y^2=4x,x=1,x=4$

$y^2=4x$

$\Rightarrow y=\pm \sqrt{4x}=\pm 2\sqrt{x}$

Area required $=2{\int }_1^{4}ydx$

$=2{\int }_1^{4}2\sqrt{x}dx$

$=4\int x^{1/2}dx$

$=4{\left[\frac{x^{\frac{3}{2}}}{\frac{3}{2}}\right]}_1^4$

$=\frac{8}{3}({\left(4\right)}^{\frac{3}{2}}-{\left(1\right)}^{\frac{3}{2}})$

$=\frac{8}{3}(8-1)$

$=\frac{56}{3}$