Question

Prove, in the following two ways, that for any integer $k$, the signed area under the graph of the function $f\left(x\right)=\sin \left(2\right(x-(\pi /4)\left)\right)$on the interval$[0,k\pi ]$is always zero:

(a) by calculating a definite integral;

(b) by considering the period and graph of the function$f\left(x\right)=\sin \left(2\right(x-(\pi /4)\left)\right)$

Short answer

Hence proved.

Step-by-step solution

Part (a) Step 1. Given information.

The given function is$f\left(x\right)=\sin \left(2\left(x-\left(\frac{\pi }{4}\right)\right)\right)$

Part(a) Step 2. Explanation.

Using definite integral,

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

Part (b) Step 1. Explanation.

The graph of the function is,

The graph shows a sine function which is half above the x-axis.