How do I dothis area prob without a calc???

Question

The area of the region bounded by the lines x=1 and y=0 and the curve y=xe^(x^2) is ???? I know the answer is (e-1)/2, but I don't know how to do this with out the calculator. Can anyone help me????

Answer 1

Let's do the indefinite integral first, then we'll plug in
the limits.
Write the integral as 1/2 * ∫ 2xe^(x^2) dx
Let u = x^2; du = 2x dx.
Then the indefinite integral is 1/2 ∫ e^u du = 1/2 e^(x²) + C.
So the definite integral is
1/2(e^1 - e^0) = (e-1)/2.