(1) The integral is straightforward; x ranges between two constants, and y ranges between two functions of x that don't intersect.
[tex]\displaystyle\int_{-2}^1\int_{-x}^{x^2+2}\mathrm dy\,\mathrm dx[/tex]
(2) First find where the two curves intersect:
y ² - 4 = -3y
y ² + 3y - 4 = 0
(y + 4) (y - 1) = 0
y = -4, y = 1 → x = 12, x = -3
That is, they intersect at the points (-3, 1) and (12, -4). Since x ranges between two explicit functions of y, you can capture the area with one integral if you integrate with respect to x first:
[tex]\displaystyle\int_{-4}^1\int_{y^2-4}^{-3y}\mathrm dx\,\mathrm dy[/tex]
(3) No special tricks here, x is again bounded between two constants and y between two explicit functions of x.
[tex]\displaystyle\int_1^5\int_0^{\frac1{x^2}}\mathrm dy\,\mathrm dx[/tex]
