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Section 15.3 : Double Integrals over General Regions

  1. Evaluate D8yx3dA where D={(x,y)|1y2,1x1+y2}
  2. Evaluate D12x2yy2dA where D={(x,y)|2x2,x2yx2}
  3. Evaluate D96y2x2dA where D is the region in the 1st quadrant bounded by y=x3 and y=4x.
  4. Evaluate D15x26ydA where D is the region bounded by x=12y2 and x=4y.
  5. Evaluate D6y(x+6)2dA where D is the region bounded by x=y2 and x=y6.
  6. Evaluate Dey2+1dA where D is the triangle with vertices (0,0), (2,4) and (8,4).
  7. Evaluate D7y3ex2+1dA where D is the region bounded byy=24x, x=9 and the x-axis.
  8. Evaluate Dx5sin(y4)dA where D is the region in the 2nd quadrant bounded by y=3x2, y=12 and the y-axis.
  9. Evaluate Dxyy2dA where D is the region shown below.
    This region consists of two triangles.  The first triangle has vertices (0,0), (2,4) and (2,-4).  The second has vertices at (0,0), (-4,4) and (-4,-4).
  10. Evaluate D12x33dA where D is the region shown below.
    This region consists of two sub regions.  The first sub region is the region in the fourth quadrant bounded by $y=x-1$, the x-axis and the y-axis.  The second sub region is the region in the second quadrant bounded by $y=1-x^{2}$, the x-axis and the y-axis.
  11. Evaluate D6y2+10yx4dA where D is the region shown below.
    This region consists of two sub regions.  The first sub region is the region in the first and fourth quadrants bounded by $x=2-y^{2}$ and the y-axis.  The second sub region is the region in the second and third quadrants bounded by $x=y^{2}-1$ and the y-axis.
  12. Evaluate Dx3y2dA where D is the region bounded by y=1x2, x=1 and y=14 in the order given below.
    1. Integrate with respect to x first and then y.
    2. Integrate with respect to y first and then x.
  13. Evaluate Dxyy3dA where D is the region bounded by y=x2, y=x2 and x=2 in the order given below.
    1. Integrate with respect to x first and then y.
    2. Integrate with respect to y first and then x.

For problems 14 – 16 evaluate the given integral by first reversing the order of integration.

  1. 802y13yx7+1dxdy
  2. 042xx23y53+1dydx
  3. 203xx5y2x3+2dydx
  4. Use a double integral to determine the area of the region bounded by x=y2 and x=y6.
  5. Use a double integral to determine the area of the region bounded by y=x2+1 and y=12x2+3.
  6. Use a double integral to determine the volume of the region that is between the xy‑plane and f(x,y)=2xy2 and is above the region in the xy-plane that is bounded by y=x2 and x=1.
  7. Use a double integral to determine the volume of the region that is between the xy‑plane and f(x,y)=1+y5x4+1 and is above the region in the xy-plane that is bounded by y=x, x=2 and the x-axis.
  8. Use a double integral to determine the volume of the region in the first octant that is below the plane given by 2x+6y+4z=8.
  9. Use a double integral to determine the volume of the region bounded by z=32y, the surface y=1x2 and the planes y=0 and z=0.
  10. Use a double integral to determine the volume of the region bounded by the planes z=42x2y, y=2x, y=0 and z=0.
  11. Use a double integral to determine the formula for the area of a right triangle with base, b and height h.
  12. Use a double integral to determine a formula for the figure below.