Question

The area of the region bounded by the curves $f\left(x\right)=\sin \pi x$ and X-axis is _______ where $xϵ[-1,2]$.

• A. $8\pi$
• B. $\frac{8}{\pi }$
• C. $\frac{6}{\pi }$
• D. $6\pi$

Answer 1

The correct option is C $\frac{6}{\pi }$

The required area under the curve $={\int }_{-1}^2\sin \pi xdx$

Put $z=\pi x⟹dx=\frac{dz}{\pi }$

$={\int }_{-\pi }^{2\pi }\sin z\frac{dz}{\pi }$

$=\frac{1}{\pi }{\int }_{-\pi }^{2\pi }\sin zdz$

$=\frac{3}{\pi }{\int }_{-\pi }^0\sin zdz$

(as the area is between curve and x-axis, the negative area is also considered from the x-axis as positive area. It can also be said that it is area of $|\sin z|$ instead of $\sin z$ and sine function is periodic with period $\pi$)

$=\frac{3}{\pi }(\cos z{)}_{-\pi }^0$

$=\frac{6}{\pi }$

This is the required solution.