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Chapter 5: Q. 87 (page 419)

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-(蟺/4)))on the interval[0,k蟺]is always zero:

(a) by calculating a definite integral;

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

Short Answer

Expert verified

Hence proved.

Step by step solution

01

Part (a) Step 1. Given information.

The given function isf(x)=sin2x-蟺4

02

Part(a) Step 2. Explanation.

Using definite integral,

鈭�0k蟺sin2x-蟺4dx=鈭�0k蟺sin2x-蟺2dx=-12cos(2k蟺)-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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