10.5: Triple Integrals over Cylindrical and Spherical Coordinates
- Page ID
- 144354
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- Use cylindrical coordinates to evaluate \(\displaystyle \iiint_R x^2\ dV\), where \(R\) is the interior of the cylinder \(x^2 + y^2 = 1\) between \(z = 0\) and \(z = 5\).
- Use cylindrical coordinates to evaluate \(\displaystyle \iiint_R (x^2 + y^2)\ dV\), where \(R\) is the interior of the sphere \(x^2 + y^2 + z^2 = 4\).
- Use spherical coordinates to evaluate \(\displaystyle \iiint_R (x^2 + y^2)\ dV\), where \(R\) is the interior of the sphere \(x^2 + y^2 + z^2 = 4\).
- Use spherical coordinates to evaluate \(\displaystyle \iiint_R yz\ dV\), where \(R\) is the region in the first octant inside \(x^2 + y^2 - 2z = 0\) and under \(x^2 + y^2 + z^2 = 4\).
- Use cylindrical coordinates or spherical coordinates to evaluate \(\displaystyle \int_0^1 \int_0^x \int_0^{\sqrt{x^2 + y^2}} \dfrac{(x^2 + y^2)^{3/2}}{x^2 + y^2 + z^2}\ dz\,dy\,dx\).
- Use cylindrical coordinates or spherical coordinates to evaluate \(\displaystyle \int_{-1}^1 \int_0^{\sqrt{1 - x^2}} \int_{\sqrt{x^2 + y^2}}^{\sqrt{2 - x^2 - y^2}} \sqrt{x^2 + y^2 + z^2}\ dz\,dy\,dx\).
- Use spherical coordinates and a triple integral to calculate the volume of the unit sphere.