# Math Help - Double Integral

1. ## Double Integral

$\int\!\!\!\!\!\int_S(x+2y)dxdy$ where S is defined by the curves: $y=2\cdot x^2$ and $y=1+x^2$.

2. Start with dy first: $2x^2 \leq y \leq 1+x^2$

To get the limits for dx: $2x^2=1+x^2 \Longleftrightarrow x^2=1$

3. Is this correct ?
$\int_{-1}^1\int_{2x^2}^{1+x^2}(x+2y)dydx=\int_{-1}^1(y^2+xy)|_{2x^2}^{x^2+1}dx=\int_{-1}^1(1+x+2x^2-x^3-3x^4)dx=$
$=(x+\frac{x^2}{2}+\frac{2x^3}{3}-\frac{x^4}{4}-\frac{3x^5}{5})|_{-1}^1=\frac{32}{15}$

4. Originally Posted by sillyme
Is this correct ?
$\int_{-1}^1\int_{2x^2}^{1+x^2}(x+2y)dydx=\int_{-1}^1(y^2+xy)|_{2x^2}^{x^2+1}dx=\int_{-1}^1(1+x+2x^2-x^3-3x^4)dx=$
$=(x+\frac{x^2}{2}+\frac{2x^3}{3}-\frac{x^4}{4}-\frac{3x^5}{5})|_{-1}^1=\frac{32}{15}$
it is correct