Question

Find the area of the region bounded by the curve y=x^2*e^x and the x-axis, 0≤x≤1

Answer 1

clearly, that's

∫[0,1] x^2 e^x dx
use integration by parts.

Answer 2

The only difficulty lies in integrating x^2 e^x
I used integration by parts twice and got
∫x^2 e^x dx = x^2 e^x - 2x e^x + 2e^x
= e^x( x^2 - 2x + 2)

so ∫ x^2 e^x dx from 0 to 1
= [e^x(x^2 - 2x + 2)] from 0 to 1
= e(1-2+2) - e^0(0-0+2)
= e - 2