10.2: Double Integrals over General Regions

1. Find the volume under the surface $z = 3xy$ and over the region $R$ bounded by $y = x^3$, $y = x^3 + 1$, $x = 0$, and $x = 1$.
    
2. Find the volume under the surface $z = xy + 1$ and over the region $R$ bounded by $x = y^2 - 1$ and $x = \sqrt{1 - y^2}$.
    
3. Find the volume bounded by $y = \sqrt{4x}$, $2x + y = 4$, $z = y$, $y = 0$, and $z = 0$.
    
4. Find the volume in the first octant bounded by $y^2 = 4 - x$ and $y = 2z$.
    
5. Find the volume in the first octant bounded by $x + y + z = 9$, $2x + 3y = 18$, and $x + 3y = 9$.
    
6. Find the volume in the first octant bounded by $x^2 + y^2 = a^2$ and $z = x + y$.
    
7. Evaluate $\displaystyle \iint_R x^2\ dA$, where $R$ is the region in the first quadrant bounded by $xy = 16$ and the lines $y = x$, $y=0$, and $x = 8$.
    
8. Evaluate $\displaystyle \iint_R \ dA$, where $R$ is the region bounded by $x = \dfrac{\pi}{2}$, $x = y - 1$, and $y = \cos x$.
    
9. Evaluate $\displaystyle \iint_R xy\ dA$, where $R$ is the triangular region with vertices $(0, 0)$, $(0, 2)$, and $(2, 2)$.
    
10. Use a double integral to find the area of region $R$, where $R$ is the triangular region with vertices $(0, 0)$, $(-3, 5)$, and $(4, 2)$.
     
11. Use a double integral to find the area of region $R$, where $R$ is the region inside the circle $(x-1)^2 + y^2 = 2$ outside the circle $x^2 + y^2 = 1$.
     
12. Reverse the order of integration for $\displaystyle \int_{-1}^{\pi/2} \int_{0}^{x+1} f(x,y)\ dy\,dx$.
     
13. Reverse the order of integration for $\displaystyle \int_{0}^{1} \int_{x-1}^{1-x} f(x,y)\ dy\,dx$.
     
14. Reverse the order of integration for $\displaystyle \int_{-1}^{0} \int_{-\sqrt{y+1}}^{\sqrt{y+1}} f(x,y)\ dx\,dy$.
     
15. Reverse the order of integration for $\displaystyle \int_{-1/2}^{1/2} \int_{-\sqrt{y^2+1}}^{\sqrt{y^2+1}} f(x,y)\ dx\,dy$.
     
16. Reverse the order of integration for $\displaystyle \int_{1}^{2} \int_{0}^{\ln x} f(x, y)\ dy\,dx$.
     
17. Reverse the order of integration for $\displaystyle \int_{0}^{1} \int_{4x}^{4} f(x, y)\ dy\,dx$.
     
18. Reverse the order of integration for $\displaystyle \int_{0}^{3} \int_{0}^{\sqrt{9 - y^2}} f(x, y)\ dx\,dy$.