Evaluate the following double integral where a = 2y \int\limits^1_0 {x} \, \int\limits^2_a {cos(x^(2) )\, dx dy\\

Question

Evaluate the following double integral where a = 2y


`\int\limits^1_0 {x} \, \int\limits^2_a {cos(x^(2) )\, dx dy\\`

Answer 1

Change the order of integration.

`\displaystyle \int_0^1 \int_(2y)^2 \cos(x^2) \, dx \, dy = \int_0^2 \int_0^(x/2) \cos(x^2) \, dy \, dx \\\\ ~~~~~~~~ = \int_0^2 \cos(x^2) y \bigg|_(y=0)^(y=x/2) \, dx \\\\ ~~~~~~~~ = \frac12 \int_0^2 x \cos(x^2) \, dx`

Substitute
`u=x^2` and
`du=2x\,dx`.

`\displaystyle \frac12 \int_0^2 x \cos(x^2) \, dx = \frac14 \int_0^4 \cos(u) \, du = \frac14 \sin(u) \bigg|_(u=0)^(u=4) = \boxed{\frac{\sin(4)}4}`