Methods Of Integration

We use the half angle identity for the trigonometric function: Thus, the integral becomes: Now, use the substitution , :

We make the substitution , so . When , ; and when , . Thus, the integral becomes:

We are given the curves and . To find the area between these curves, we first determine the points of intersection. Set: Solving this: Thus, the curves intersect at and . The area between the curves can be found by integrating the difference between the upper function ( ) and the lower function ( ) over the interval . The area is: Breaking this into two integrals: First, calculate the integral of : Now, calculate the integral of : Subtracting the two integrals: Thus, the area of the region is square units.
\begin{figure}[ht] \centering \includegraphics[width=0.7\textwidth]{ig4.png} \end{figure}