Unbeatable area

Calculus Level 3

Let the area bounded by the function f ( x ) = max ⁡ ( sin ⁡ ( x ) , cos ⁡ ( x ) ) f(x) = \max(\sin(x), \cos(x)) and the x x -axis on the interval 0 ≤ x ≤ π 2 0 \leq x\leq \frac\pi2 be k k , find the value of k 4 k^4 .


The answer is 4.

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Atul Shivam by Atul Shivam , 23, India

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2 solutions

Kiran Yarlagadda
Oct 2, 2015

in the region 0<x<pi/4 cos(x) is maximum,while in the region pi/4<x<pi/2 sin(x) is max. int(cos(x),0,pi/4)=1/sqrt(2)--->1; int(sin(x),pi/4,pi/2)=1/sqrt(2)--->2; 1+2=(sqrt(2)); sqrt(2).^4=4

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Aditya Kumar
Oct 3, 2015

Integrate the given function from 0 to pi/2 Then break it into intervals 0to pi/4 and pi/4 to pi/2 It comes as [sin x] 0 to pi/4 + [cos x] pi/4 to pi/2 Therefore solving we get 1/root2 +1/root2 =root2= k Now k^4=[root2]^4 = 4

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