Question

Find the area bounded by the 1 arc curve y=sin a x and the x axis

Answer 1

$\textbf{Given:}$

$\text{Equation of the curve is y=\sin{ax} }$

$\textbf{To find:}$

$\text{The area bounded y the one arc of the curve and x axis}$

$\textbf{Solution:}$

$\textbf{First we find  the points where the curve meets the x axis}$

$\text{Put y=0 }$

$\sin{ax}=0$

$\implies\,ax=n\,\pi$

$\implies\,x=\dfrac{n\,\pi}{a},\;\;n{\in}Z$

$\implies\,x=0,\dfrac{\pi}{a},\dfrac{2\,\pi}{a}........$

$\text{The limits for one arc of the curve are x=0 and x=\dfrac{\pi}{a} }$

$\textbf{Area bounded by one arc of the arc}$

$=\int\limits_{a}^{b}\,y\,dx$

$=\int\limits_{0}^{\frac{\pi}{a}}\,\sin{ax}\,dx$

$=[\dfrac{-\cos{ax}}{a}]\limits_{0}^{\frac{\pi}{a}}$

$=\dfrac{-1}{a}[\cos{ax}]\limits_{0}^{\frac{\pi}{a}}$

$=\dfrac{-1}{a}[\cos{a(\frac{\pi}{a})}-\cos{0}]$

$=\dfrac{-1}{a}[\cos{\pi}-\cos{0}]$

$=\dfrac{-1}{a}[-1-1]$

$=\dfrac{-1}{a}[-2]$

$=\dfrac{2}{a}\;\text{square units}$

$\textbf{Answer:}$

$\textbf{Area bounded by one arc of the curve and x axis is \bf\dfrac{2}{a} square units}$