How to find the area of the region bounded by the graphs y = x^3 + 2x^2 and the line y=x+1

Question

1) Define the intersection points. (Show how you get the points).

2) Shaded the region bounded.

3) Evaluate the area (show the calculations).

Answer 1

The intersection is found by equating the two curves f₁(x) = f₂(x) or x³+2x²=x+1 or x³+2x²-x-1 = 0 There are three points of intersection which makes the solution problematic because the function switch places relative to each other on in the y direction. First, we find the roots of the equation listed. There is a cubic formula. but read on if you are moved to use it. I don't see an obvious root, so I would do this numerically. You can plot both functions on desmos or a graphing calculator and get approximations to the intersection points by just plotting the two curves and using trace. Honestly, the exact solution of roots for this equation I get from Wolfram Alpha are seriously gross, making me think that the curve is incorrect or you are supposed to approximate the roots

In order to find the positive area between the curves (no domain limit given), we have to break the integral into two pieces because the curves switch places relative to each other:

Integral from low pt to middle point of (f₂ - f₁)dx + Integral from the middle point to high point of (f₁-f₂)dx

Where the limits of the integrals are the intersection points in order from lowest to highest.

(If it was x² instead of 2x² you'd have 1,-1 as roots...)