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

(a) by calculating a definite integral;

(b) by considering the period and graph of the functionf(x)=sin(2(x-(π/4)))

Short Answer

Expert verified

Hence proved.

Step by step solution

01

Part (a) Step 1. Given information.

The given function isf(x)=sin2x-π4

02

Part(a) Step 2. Explanation.

Using definite integral,

0kπsin2x-π4dx=0kπsin2x-π2dx=-12cos(2kπ)-12cos(0)=-12+12=0

03

Part (b) Step 1. Explanation.

The graph of the function is,

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

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