Integral Area

Calculus Level pending

Find the measure of the area of the region bounded by the graphs of the curves of Equations y = x 2 y = x^2 and y = x + 6 y = x +6

Details :

  • Can Be Written as a b \frac{a}{b} Where a a and b b are coprime positive integers.

Find a + b a+b


The answer is 131.

Gabriel Merces by Gabriel Merces , 23, Brazil

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1 solution

Tom Engelsman
May 14, 2021

The line the parabola intersect in the points x 2 = x + 6 ⇒ x 2 − x − 6 = ( x − 3 ) ( x + 2 ) = 0 ⇒ x = − 2 , 3. x^2=x+6 \Rightarrow x^2-x-6= (x-3)(x+2)=0 \Rightarrow x = -2, 3. The required area computes to:

A = ∫ − 2 3 x + 6 − x 2 d x = x 2 2 + 6 x − x 3 3 ∣ − 2 3 = 125 6 . \large A = \int_{-2}^{3} x+6-x^2 dx = \frac{x^2}{2} + 6x - \frac{x^3}{3}|_{-2}^{3} = \boxed{\frac{125}{6}}.

Thus, a + b = 125 + 6 = 131 . a+b=125+6=\boxed{131}.

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