2. Find the area of the region bounded by the curve y=x^3-2x^2-5x+6, the x-axis and the lines x=-1 and x=2

Answer 1

Find the integral from -1 to 2 of (x³-2x²-5x+6)dx to be 12.25

Answer 2

The key to this problem is recognizing that f(x) = (x-1)(x-3)(x+2) (plug in -2, 1, and 3 into the polynomial to verify).

Thus, on the interval defined, f(x) >0 on (-1, 1) and negative on (1, 2)

We break the integral into 2 parts.

Using I[a, b] for the integral from a to b and E[a, b] for the evaluation from a to b,

I[-1, 1] x^3 - 2 x^2 - 5x + 6 dx - I[-1, 1] x^3 - 2 x^2 - 5x + 6 dx =

x^4/4 - 2/3 x^3 - 5/2 x^2 + 6x E[-1, 1] - x^4/4 - 2/3 x^3 - 5/2 x^2 + 6x E[1, 2] =

1/4 - 2/3 - 5/2 + 6 - (1/4 + 2/3 - 5/2 - 6) -(4 - 16/3 - 10 + 12 - (1/4 - 2/3 - 5/2 + 6)) =

13 1/12