Find the area of the region bounded by the curves y=x^3 and y= -4x^2 The area is

Question

Find the area of the region bounded by the curves y=x^3 and y= -4x^2
The area is? (simplify answer show an integer or a fraction)

Please show work. Thanks

Explanation / Answer

The area can be found by the integral of the difference of the two functions between their points of intersection. So begin by finding where they intersect: (x - 3)^2 = 16 x - 3 = ±4 x = 3 ± 4 x = -1 or 7 So the limits of integration will be from x = -1 to x = 7. Next, we must determine which function is the "top function" (ie is greater). By substituting any value of x in the interval (-1, 7), we can see that 16 > (x - 3)^2 on (-1, 7). So we'll subtract (x - 3)^2 from 16 and integrate that on the interval (-1, 7) to find the area: A = (-1 to 7)? 16 - (x - 3)^2 dx = (-1 to 7)? 16 - (x^2 - 6x + 9) dx = (-1 to 7)? 16 - x^2 + 6x - 9 dx = (-1 to 7)? 7 - x^2 + 6x dx = [7x - (1/3)x^3 + 3x^2](-1 to 7) = [7(7) - (1/3)(7)^3 + 3(7)^2] - [7(-1) - (1/3)(-1)^3 + 3(-1)^2] = 245/3 - (-11/3) = 256/3 So the answer is d) 256/3.

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