Integral and Series Problems
Polar Coordinate and Double Integral
  1. Find the volume of a solid with height 1 and base a region bounded by y=x(6-x) and y=4x-8 in the first quadrant.
  2. Find the volume of the solid below z=xy over the region bounded by y^2=x+1 and y=1-x.
  3. Evaluate each of the following:

    (1)   \begin{equation*} \int _0 ^1 \int _{\arcsin y} ^{\pi /2} e^{\cos x} dx dy \end{equation*}

     

    (2)   \begin{equation*} \int _0 ^2 \int _{0} ^{3} \frac{y^2}{(1+xy^2)^3} dx dy \end{equation*}

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