Practices on Double Integrals

  1. Find the double integrals under the function $f(x,y)$ and the region $R.$

    1. $f(x,y)=2x+2y-1,$ $R$ is bounded by $x=y,x+y-4=0$ and $x=0.$

    2. $f(x,y)=2xy-3x^{2},$ $R$ is bounded by $y=\log _{e}x,$ MATH and $y=-2.$

    3. MATH $R$ is the circle of $x^{2}+y^{2}=2$ in the first quadrant.

    4. MATH $R$ is bounded by $x+y=2,$ $x-2y=2,$ $y=0,$and $y=2.$

  2. Find the volume under $z=2x+y+10$ and above MATH and $xy$ plane.

  3. Find the volume bounded by the surface $y^{2}+z^{2}=4,$ plane: $y=x,$ and the $xy$ plane in the first octant. Use Maple to sketch the intersections.

  4. Find the volume bounded by the surface $z=16-x^{2}-4y^{2}$ in the first octant. Use Maple to sketch the intersections

  5. Set up the double integral for the volume of the ellipsoid MATH.

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