Double and triple integrals

Compute the double integral

$${\iint }_{R}y\sin xdxdy,R=[0,\pi ]\times [0,1].$$

Solution available to those who have unlocked the subject.

Compute the double integral

$${\iint }_{R}x{e}^{xy}dxdy,R=[0,1]\times [0,2].$$

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Compute the double integral

$${\iint }_{T}xdxdy,$$

where $T$ is the triangle with vertices $(0,0)$, $(1,0)$ and $(1,1)$.

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Compute the double integral

$${\iint }_D(x+y)dxdy,D=\{(x,y):0\le x\le 1,0\le y\le x^2\}.$$

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Compute the iterated integral

$${\int }_0^1\left({\int }_x^1\cos (y^2)dy\right)dx.$$

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Compute the double integral

$${\iint }_D\frac{1}{1+x^2+y^2}dxdy,D=\{(x,y):x^2+y^2\le 1\}.$$

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Find the centroid of the homogeneous lamina

$$D=\{(x,y):x^2+y^2\le 1,x\ge 0,y\ge 0\}.$$

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Compute the area of the region in the first quadrant bounded by the hyperbolas $xy=1$ and $xy=3$ and by the lines $y=x$ and $y=2x$.

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Compute the double integral

$${\iint }_D\left(\frac{x^2}{9}+\frac{y^2}{4}\right)dxdy,D=\left\{(x,y):\frac{x^2}{9}+\frac{y^2}{4}\le 1\right\}.$$

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Compute the double integral

$${\iint }_D\left(x^3+5y+2\right)dxdy,D=\{(x,y):x^2+y^2\le 4\}.$$

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Compute the triple integral

$${\iiint }_{T}zdxdydz,$$

where $T$ is the tetrahedron bounded by the coordinate planes and by the plane $x+y+z=2$.

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Compute the triple integral

$${\iiint }_{E}zdxdydz,E=\{(x,y,z):x^2+y^2\le z\le 4\}.$$

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Compute the triple integral

$${\iiint }_{E}z\sqrt{x^2+y^2}dxdydz,E=\{(x,y,z):x^2+y^2\le 1,0\le z\le 2\}.$$

Solution available to those who have unlocked the subject.

Compute the triple integral

$${\iiint }_E\left(x^2+y^2+z^2\right)dxdydz,E=\{(x,y,z):x^2+y^2+z^2\le 4\}.$$

Solution available to those who have unlocked the subject.