Q. 63 - Calculus

Question

Use the Fundamental Theorem of Calculus to find the exact

values of each of the definite integrals in Exercises $19–64$. Use

a graph to check your answer.

${\int }_0^{5\pi /4}\left|\sin x\right|dx$

Step-by-Step Solution

Answer

The value of integral is $3-\frac{1}{\sqrt{2}}$ and the plot is

Step 1. Given information

An expression is given as ${\int }_0^{5\pi /4}\left|\sin x\right|dx$

Step 2. Evaluating integral

Simplify the integral as,

$$\begin{gathered} {\int }_0^{5\frac{\pi }{4}}\left|\sin x\right|dx \\ ={\int }_0^{\frac{\pi }{2}}\sin xdx+{\int }_{\frac{\pi }{2}}^{\pi }-\sin xdx+{\int }_{\pi }^{\frac{5\pi }{4}}\sin xdx \\ =\left[-\cos \left(\frac{\pi }{2}\right)+\cos \left(0\right)\right]+\left[-\cos \left(\pi \right)+\cos \left(\frac{\pi }{2}\right)\right]+\left[-\cos \left(\frac{5\pi }{4}\right)-\cos \left(\pi \right)\right] \\ =0+1+1+0-\frac{1}{\sqrt{2}}+1 \\ =3-\frac{1}{\sqrt{2}} \end{gathered}$$

And the plot is