Double Integrals over General Regions

1. Evaluate  \(\displaystyle{ \iint_D (2x+6y^2)\, dA} \), where \( D \) is the region bounded by the parabola \(y=x^2\) and  the line \(y=1\).   solution

2. Evaluate  \(\displaystyle{ \iint_D x \cos y \, dA} \), where \( D \) is the region bounded by \(y=0, \: y=x^2\), and  \(x=2\).   solution

3. Evaluate  \(\displaystyle{ \iint_D x^2 e^{xy} \, dA} \), where \( D \) is the triangle with vertices \((0,0), (2,1) \) and \((0,1)\).   solution

4. Find the volume of the solid that lies under the elliptic paraboloid \( \displaystyle{f(x,y) = 3x^2+y^2}\) and above the region \(D\) bounded by \( x = y \) and \( x = y^2-2 \).   solution

5. Evaluate the integral \(\displaystyle{\int_0^1\int_{2y}^2 e^{x^2} \, dx \, dy} \)      solution

6. Evaluate the integral \(\displaystyle{\int_0^{1/4} \int_{\sqrt{x}}^{1/2}\frac{e^y}{y} \, dy \, dx} \)     solution

7.  Sketch the region of integration and reverse the order of integration for the following double integral.   solution
\[ \int_0^{27} \int_{\sqrt[3]{x}}^2 f(x,y) \, dy\,dx \]