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Question

Evaluateab(x-a)(b-x)dx


A

π(b-a)28

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B

π(b+a)28

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C

b-a2

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D

b+a2

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Solution

The correct option is A

π(b-a)28


Explanation for correct option:

Integrating the given integration:

Let I=ab(x-a)(b-x)dx

Substitute x=acos2θ+bsin2θ

dx=(-2acosθsinθ+2bsinθcosθ)dθddθsinθ=cosθ,ddθcosθ=-sinθdx=(-asin2θ+bsin2θ)dθ[sin2θ=2sinθcosθ]dx=(b-a)sin2θdθ

I=0π2(b-a)sin2θ×(b-a)cos2θ(b-a)sin2θdθ=0π2(b-a)sinθcosθ(b-a)sin2θdθ=0π2(b-a)2×22sinθcosθ×sin2θdθ=0π2(b-a)2×12sin2θ×sin2θdθ=0π2(b-a)2×12sin22θdθ=(b-a)220π2sin22θdθ=(b-a)220π21-cos4θ2dθcos2θ=1-2sin2θ2sin22θ=1-cos4θ=(b-a)24θ+sin4θ40π2dθ=(b-a)24π2+sin4π24-0-0=(b-a)24π2+0-0-0=π(b-a)28

Thus, ab(x-a)(b-x)dx=π(b-a)28

Hence, option (A) is the correct answer


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