Question

Reverse the order of integration and evaluate the following integral:

∫ from 0 to 1 ∫ from 0 to x^2 (x + y) dy dx
a) 1/6
b) 1/4
c) 1/3
d) 1/2

Answer 1

To reverse the order of integration, integrate (x + y) with respect to y from 0 to x² and then integrate x³/2 with respect to x from 0 to 1. The value of the integral is 1/2.

Step-by-step explanation:

To reverse the order of integration, we need to change the limits of integration and swap the order of the variables. The given integral is ∫01 ∫0x² (x + y) dy dx. To reverse the order, we integrate with respect to y first and then with respect to x.

First, integrate (x + y) with respect to y from 0 to x²:
∫(x + y) dy = xy + ½y² |y=0x² = x³/2

Then, integrate x³/2 with respect to x from 0 to 1:
∫(x³/2) dx = ½x⁴ |x=01 = ½ * 1⁴ - ½ * 0⁴ = ½

Therefore, the value of the integral is ½. The correct option is d) 1/2.