Nasty areas!

Calculus Level 5

If the area bounded by the curves y = x 2 y = x^2 , y = ∣ 2 − x 2 ∣ y= |2-x^2| and y = 2 y=2 , which lies to the right of the line x = 1 x=1 can be expressed as a − b c d \dfrac{a - b \sqrt{c}}{d} , then what is a + b + c + d a+b+c+d ?


The answer is 37.

Abhishek Singh by Abhishek Singh , 24, India

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1 solution

Kishan Jani
Apr 25, 2019

Graphing the functions, The total area A A would be the sum of the areas A 1 A_{1} and A 2 A_{2}

A 1 = ∫ 1 2 x 2 − ( 2 − x 2 ) d x = 4 − 2 2 3 A_{1}=\int_{1}^{\sqrt{2}}x^{2}-(2-x^{2})dx=\frac{4-2\sqrt{2}}{3}

A 2 = ∫ 2 2 2 − ( x 2 − 2 ) d x = 16 − 10 2 3 A_{2}=\int_{\sqrt{2}}^{2}2-(x^{2}-2)dx=\frac{16-10\sqrt{2}}{3}

Adding these, A = 20 − 12 2 3 A=\frac{20-12\sqrt{2}}{3}

a + b + c + d = 37 a+b+c+d=37

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