General Notice
I have been informed that on March 7th from 6:00am to 6:00pm Central Time Lamar University will be doing some maintenance to replace a faulty UPS component and to do this they will be completely powering down their data center.
Unfortunately, this means that the site will be down during this time. I apologize for any inconvenience this might cause.
Paul
February 18, 2026
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Section 15.7 : Triple Integrals in Spherical Coordinates
- Evaluate \( \displaystyle \iiint\limits_{E}{{10xz + 3\,dV}}\) where \(E\) is the region portion of \({x^2} + {y^2} + {z^2} = 16\) with \(z \ge 0\). Solution
- Evaluate \( \displaystyle \iiint\limits_{E}{{{x^2} + {y^2}\,dV}}\) where \(E\) is the region portion of \({x^2} + {y^2} + {z^2} = 4\) with \(y \ge 0\). Solution
- Evaluate \( \displaystyle \iiint\limits_{E}{{3z\,dV}}\) where \(E\) is the region inside both \({x^2} + {y^2} + {z^2} = 1\) and \(z = \sqrt {{x^2} + {y^2}} \). Solution
- Evaluate \( \displaystyle \iiint\limits_{E}{{{x^2}\,dV}}\) where \(E\) is the region inside both \({x^2} + {y^2} + {z^2} = 36\) and \(z = - \sqrt {3{x^2} + 3{y^2}} \). Solution
- Evaluate the following integral by first converting to an integral in spherical coordinates. \[\int_{{ - 1}}^{0}{{\int_{{ - \sqrt {1 - {x^{\,2}}} }}^{{\sqrt {1 - {x^{\,2}}} }}{{\int_{{\sqrt {6{x^{\,2}} + 6{y^{\,2}}} }}^{{\sqrt {7 - {x^{\,2}} - {y^{\,2}}} }}{{\,\,\,18y\,\,\,dz}}\,dy}}\,dx}}\] Solution