Double Integrals Over General Regions

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  1. Evaluate the iterated integral 010x2x+2ydydx .
  2. Evaluate the double integral DxcosydA if D is bounded by y=0 , y=x2 , and x=1 .
  3. Evaluate the double integral Dy3dA if D is a triangle with vertices 02 , 11 , and 32 .
  4. Find the volume of the solid under the plane x+2yz=0 and above the region bounded by y=x and y=x4 .
  5. Find the volume of the solid bounded by the planes x=0 , y=0 , z=0 , and x+y+z=1 .
  6. Find the volume of the solid bounded by the cylinder x2+y2=1 and the planes y=z , x=0 , and z=0 in the first octant.
  7. Sketch the region of integration and change the order of integration for

    120lnxfxydydx .

    0ln2ey2fxydxdy .
  8. Evaluate the integral


    by reversing the order of integration.

  9. Express D as a union of Type 1 and/or Type 2 regions and evaluate the integral Dx2dA . The curve plotted in the figure is y=1x2 .

    composite integration region

  10. The double integral over a region D can be written as a sum of iterated integrals

    DfxydA=0102yfxydxdy+1303yfxydxdy .

    Sketch the region D and express the double integral as an iterated integral with reversed order of integration.