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Section 4-4 : Double Integrals in Polar Coordinates

  1. Evaluate \( \displaystyle \iint\limits_{D}{{3x{y^2} - 2\,dA}}\) where \(D\) is the unit circle centered at the origin.
  2. Evaluate \( \displaystyle \iint\limits_{D}{{4x - 2y\,dA}}\) where \(D\) is the top half of region between \({x^2} + {y^2} = 4\) and \({x^2} + {y^2} = 25\).
  3. 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.
  4. 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 \({x^2} + {y^2} = 7\).
  5. 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 \({x^2} + {y^2} = 36\).
  6. Use a double integral to determine the area of the region that is inside \(r = 6 - 4\cos \theta \).
  7. Use a double integral to determine the area of the region that is inside \(r = 4\) and outside \(r = 8 + 6\sin \theta \).
  8. Evaluate the following integral by first converting to an integral in polar coordinates. \[\int_{{ - 2}}^{0}{{\int_{{ - \sqrt {4 - {y^{\,2}}} }}^{{\sqrt {4 - {y^{\,2}}} }}{{\,\,\,{x^2}\,dx}}\,dy}}\]
  9. Evaluate the following integral by first converting to an integral in polar coordinates. \[\int_{{ - 1}}^{1}{{\int_{0}^{{\sqrt {1 - {x^{\,2}}} }}{{\,\,\,\sqrt {{x^2} + {y^2}} \,dy}}\,dx}}\]
  10. 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.
  11. Use a double integral to determine the volume of the solid that is bounded by \(z = 12 - 3{x^2} - 3{y^2}\) and \(z = {x^2} + {y^2} - 8\).
  12. 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 \({x^2} + {y^2} + {z^2} = 16\).
  13. Use a double integral to derive the area of a circle of radius \(a\).
  14. 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.
    The region is the portion of a ring in the first quadrant.  The inner radius of the ring is given as “a” and the outer radius of the ring is given as “b”.  The ring starts at a line that makes an angle of $\alpah$ with the positive x-axis and ends at a line that makes an angle of $\beta$ with the positive x-axis where $\alpha < $\beta$.