13 - 22 evaluate the triple integral.\n13. $iiint_e ydv$, where $e={(x,y,z)mid0leq xleq3,0leq yleq x,x…

13 - 22 evaluate the triple integral.\n13. $iiint_e ydv$, where $e={(x,y,z)mid0leq xleq3,0leq yleq x,x - yleq zleq x + y}$\n14. $iiint_e e^{z/y}dv$, where $e={(x,y,z)mid0leq yleq1,yleq xleq1,0leq zleq xy}$\n15. $iiint_e(1/x^{3})dv$, where $e={(x,y,z)mid0leq yleq1,0leq zleq y^{2},1leq xleq z + 1}$\n16. $iiint_esin ydv$, where $e$ lies below the plane $z = x$ and above the triangular region with vertices $(0,0,0),(pi,0,0)$, and $(0,pi,0)$\n17. $iiint_e 6xydv$, where $e$ lies under the plane $z = 1 + x + y$ and above the region in the $xy$-plane bounded by the curves $y=sqrt{x},y = 0$, and $x = 1$\n18. $iiint_e(x - y)dv$, where $e$ is enclosed by the surfaces $z=x^{2}-1,z = 1 - x^{2},y = 0$, and $y = 2$\n19. $iiint_t y^{2}dv$, where $t$ is the solid tetrahedron with vertices $(0,0,0),(2,0,0),(0,2,0)$, and $(0,0,2)$\n20. $iiint_t xzdv$, where $t$ is the solid tetrahedron with vertices $(0,0,0),(1,0,1),(0,1,1)$, and $(0,0,1)$\n21. $iiint_e xdv$, where $e$ is bounded by the paraboloid $x = 4y^{2}+4z^{2}$ and the plane $x = 4$\n22. $iiint_e zdv$, where $e$ is bounded by the cylinder $y^{2}+z^{2}=9$ and the planes $x = 0,y = 3x$, and $z = 0$ in the first octant\n23 - 26 use a triple integral to find the volume of the given solid.\n23. the tetrahedron enclosed by the coordinate planes and the plane $2x + y+z = 4$\n24. the solid enclosed by the paraboloids $y=x^{2}+z^{2}$ and $y = 8 - x^{2}-z^{2}$

13 - 22 evaluate the triple integral.\n13. $iiint_e ydv$, where $e={(x,y,z)mid0leq xleq3,0leq yleq x,x - yleq zleq x + y}$\n14. $iiint_e e^{z/y}dv$, where $e={(x,y,z)mid0leq yleq1,yleq xleq1,0leq zleq xy}$\n15. $iiint_e(1/x^{3})dv$, where $e={(x,y,z)mid0leq yleq1,0leq zleq y^{2},1leq xleq z + 1}$\n16. $iiint_esin ydv$, where $e$ lies below the plane $z = x$ and above the triangular region with vertices $(0,0,0),(pi,0,0)$, and $(0,pi,0)$\n17. $iiint_e 6xydv$, where $e$ lies under the plane $z = 1 + x + y$ and above the region in the $xy$-plane bounded by the curves $y=sqrt{x},y = 0$, and $x = 1$\n18. $iiint_e(x - y)dv$, where $e$ is enclosed by the surfaces $z=x^{2}-1,z = 1 - x^{2},y = 0$, and $y = 2$\n19. $iiint_t y^{2}dv$, where $t$ is the solid tetrahedron with vertices $(0,0,0),(2,0,0),(0,2,0)$, and $(0,0,2)$\n20. $iiint_t xzdv$, where $t$ is the solid tetrahedron with vertices $(0,0,0),(1,0,1),(0,1,1)$, and $(0,0,1)$\n21. $iiint_e xdv$, where $e$ is bounded by the paraboloid $x = 4y^{2}+4z^{2}$ and the plane $x = 4$\n22. $iiint_e zdv$, where $e$ is bounded by the cylinder $y^{2}+z^{2}=9$ and the planes $x = 0,y = 3x$, and $z = 0$ in the first octant\n23 - 26 use a triple integral to find the volume of the given solid.\n23. the tetrahedron enclosed by the coordinate planes and the plane $2x + y+z = 4$\n24. the solid enclosed by the paraboloids $y=x^{2}+z^{2}$ and $y = 8 - x^{2}-z^{2}$

Answer

Explanation:

Step1: Identify the limits of integration for problem 13

For the triple - integral $\iiint_E y\ dV$ where $E={(x,y,z)\mid0\leq x\leq3,0\leq y\leq x,x - y\leq z\leq x + y}$, the triple - integral can be written as $\int_{0}^{3}\int_{0}^{x}\int_{x - y}^{x + y}y\ dz\ dy\ dx$.

Step2: Integrate with respect to $z$ first

$\int_{0}^{3}\int_{0}^{x}y\left[z\right]{z = x - y}^{z = x + y}dy\ dx=\int{0}^{3}\int_{0}^{x}y((x + y)-(x - y))dy\ dx=\int_{0}^{3}\int_{0}^{x}y(2y)dy\ dx=\int_{0}^{3}\int_{0}^{x}2y^{2}dy\ dx$.

Step3: Integrate with respect to $y$

$\int_{0}^{3}2\left[\frac{y^{3}}{3}\right]{y = 0}^{y = x}dx=\int{0}^{3}\frac{2}{3}x^{3}dx$.

Step4: Integrate with respect to $x$

$\frac{2}{3}\left[\frac{x^{4}}{4}\right]_{0}^{3}=\frac{2}{3}\times\frac{3^{4}}{4}=\frac{2\times81}{12}=\frac{27}{2}$.

Answer:

$\frac{27}{2}$