Find the area between the curves fx x1 ex2 2x and gx x2x3

Solution

Area under f(x) = integ f(x) dx from 0 to 1 integ (x+1) e^(x^2+2x) dx let x^2 + 2x = t => 2(x+1) dx = dt => (x+1)dx = dt/2 and t goes from 0 to 3 integ e^t/2 dt from 0 to 3 = e^t/2 from 0 to 3 = e^3/2 - 1/2 = (e^3-1)/2 Area under g(x) = integ x^2/(x^3+1) dx from 0 to 1 x^3 + 1 = t => 3x^2 dx = dt => x^2 dx = dt/3 t goes from 1 to 2 integ 1/3t dt from 1 to 2 1/3 ln t from 1 to 2 = 1/3 (ln 2 - ln 1) = 1/3 * (ln 2) Area between f(x) and g(x) = (e^3-1)/2 - (ln 2)/3 = 9.312