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If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
Section 15.4 : Double Integrals in Polar Coordinates
- Evaluate \( \displaystyle \iint\limits_{D}{{3x{y^2} - 2\,dA}}\) where \(D\) is the unit circle centered at the origin.
- Evaluate \( \displaystyle \iint\limits_{D}{{4x - 2y\,dA}}\) where \(D\) is the top half of region between \({x^2} + {y^2} = 4\) and \({x^2} + {y^2} = 25\).
- Evaluate \( \displaystyle \iint\limits_{D}{{6xy + 4{x^2}\,dA}}\) where \(D\) is the portion of \({x^2} + {y^2} = 9\) in the 2nd quadrant.
- Evaluate \( \displaystyle \iint\limits_{D}{{\sin \left( {3{x^2} + 3{y^2}} \right)\,dA}}\) where \(D\) is the region between \({x^2} + {y^2} = 1\) and \({x^2} + {y^2} = 7\).
- Evaluate \( \displaystyle \iint\limits_{D}{{{{\bf{e}}^{1 - {x^{\,2}} - {y^{\,2}}}}\,dA}}\) where \(D\) is the region in the 4th quadrant between \({x^2} + {y^2} = 16\) and \({x^2} + {y^2} = 36\).
- Use a double integral to determine the area of the region that is inside \(r = 6 - 4\cos \theta \).
- Use a double integral to determine the area of the region that is inside \(r = 4\) and outside \(r = 8 + 6\sin \theta \).
- Evaluate the following integral by first converting to an integral in polar coordinates. \[\int_{{ - 2}}^{0}{{\int_{{ - \sqrt {4 - {y^{\,2}}} }}^{{\sqrt {4 - {y^{\,2}}} }}{{\,\,\,{x^2}\,dx}}\,dy}}\]
- Evaluate the following integral by first converting to an integral in polar coordinates. \[\int_{{ - 1}}^{1}{{\int_{0}^{{\sqrt {1 - {x^{\,2}}} }}{{\,\,\,\sqrt {{x^2} + {y^2}} \,dy}}\,dx}}\]
- Use a double integral to determine the volume of the solid that is below \(z = 9 - 4{x^2} - 4{y^2}\) and above the \(xy\)-plane.
- Use a double integral to determine the volume of the solid that is bounded by \(z = 12 - 3{x^2} - 3{y^2}\) and \(z = {x^2} + {y^2} - 8\).
- Use a double integral to determine the volume of the solid that is inside both the cylinder \({x^2} + {y^2} = 9\) and the sphere \({x^2} + {y^2} + {z^2} = 16\).
- Use a double integral to derive the area of a circle of radius \(a\).
- Use a double integral to derive the area of the region between circles of radius a and b with \(\alpha \le \theta \le \beta \). See the image below for a sketch of the region.
