Section 15.3 : Double Integrals over General Regions

1. Evaluate$\displaystyle \iint\limits_{D}{{42{y^2} - 12x\,dA}}$ where $D = \left\{ {\left( {x,y} \right)|0 \le x \le 4,{{\left( {x - 2} \right)}^2} \le y \le 6} \right\}$ Solution
2. Evaluate $\displaystyle \iint\limits_{D}{{2y{x^2} + 9{y^3}\,dA}}$ where $D$ is the region bounded by $\displaystyle y = \frac{2}{3}x$ and $y = 2\sqrt x$. Solution
3. Evaluate $\displaystyle \iint\limits_{D}{{10{x^2}{y^3} - 6\,dA}}$ where $D$ is the region bounded by $x = - 2{y^2}$ and $x = {y^3}$. Solution
4. Evaluate $\displaystyle \iint\limits_{D}{{x\left( {y - 1} \right)\,dA}}$ where $D$ is the region bounded by $y = 1 - {x^2}$ and $y = {x^2} - 3$. Solution
5. Evaluate $\displaystyle \iint\limits_{D}{{5{x^3}\cos \left( {{y^3}} \right)\,dA}}$ where $D$ is the region bounded by $y = 2$, $\displaystyle y = \frac{1}{4}{x^2}$ and the $y$-axis. Solution
6. Evaluate $\displaystyle \iint\limits_{D}{{\frac{1}{{{y^{\frac{1}{3}}}\left( {{x^3} + 1} \right)}}\,dA}}$ where $D$ is the region bounded by $x = - {y^{\frac{1}{3}}}$, $x = 3$ and the $x$-axis. Solution
7. Evaluate $\displaystyle \iint\limits_{D}{{3 - 6xy\,dA}}$ where $D$ is the region shown below.
   
   Solution
8. Evaluate $\iint\limits_{D}{{{{\bf{e}}^{\,{y^{\,4}}}}\,dA}}$ where $D$ is the region shown below.
   
   Solution
9. Evaluate $\displaystyle \iint\limits_{D}{{7{x^2} + 14y\,dA}}$ where $D$ is the region bounded by $x = 2{y^2}$ and $x = 8$ in the order given below.
   1. Integrate with respect to $x$ first and then $y$.
   2. Integrate with respect to $y$ first and then $x$.
   Solution

For problems 10 & 11 evaluate the given integral by first reversing the order of integration.

1. $\displaystyle \int_{0}^{3}{{\int_{{2x}}^{6}{{\sqrt {{y^2} + 2} \,dy}}\,dx}}$ Solution
2. $\displaystyle \int_{0}^{1}{{\int_{{ - \sqrt y }}^{{{y^{\,2}}}}{{6x - y\,dx}}\,dy}}$ Solution
3. Use a double integral to determine the area of the region bounded by $y = 1 - {x^2}$ and $y = {x^2} - 3$. Solution
4. Use a double integral to determine the volume of the region that is between the $xy$‑plane and$f\left( {x,y} \right) = 2 + \cos \left( {{x^2}} \right)$ and is above the triangle with vertices $\left( {0,0} \right)$, $\left( {6,0} \right)$ and $\left( {6,2} \right)$. Solution
5. Use a double integral to determine the volume of the region bounded by $z = 6 - 5{x^2}$ and the planes $y = 2x$, $y = 2$,$x = 0$ and the $xy$-plane. Solution
6. Use a double integral to determine the volume of the region formed by the intersection of the two cylinders ${x^2} + {y^2} = 4$ and ${x^2} + {z^2} = 4$. Solution