Mathematical Scratchpad

Double Integral Example Worksheet

Double Integrals over general regions in $x,y$ coordinates

Sketch regions too

  1. MATH

    Inner: MATH

    Completion: MATH

  2. MATH where $D$ is the triangle with vertices MATH

    MATH

  3. MATH where $D$ is the region bounded by $y=x^{3}$ and $y=\sqrt{x}$

    MATH

Reverse order of integration.

  1. MATH

  2. MATH

  3. MATH

  4. MATH

Find Volume of solid

  1. Tetrahedron in first octant bounded by coordinate planes and $z=7-3x-2y.$

    MATH

  2. Solid inside both the sphere MATH and paraboloid $2z=x^{2}+y^{2}.$

    MATH

Double Integrals using polar coordinates

Direct Computations in polar coordinates

  1. Compute MATH

    Inner: MATH Using $u=-r^{2}$ and $du=-2r~dr$

    Completion: MATH

  2. FInd the area bounded by the cardioid MATH
    DoubleIntegral_Worksheet_Key__29.png

    MATH

  3. Find the area bounded by one leaf of the rose $r=4\cos \theta $
    DoubleIntegral_Worksheet_Key__32.png

    MATH

  4. Find area inside both $r=1$ and $r=2\sin \theta .$
    DoubleIntegral_Worksheet_Key__36.png

    MATH

Convert from Cartesian ( $x,y$) to polar coordinates before integrating

  1. Find MATH where $D$ is the region bounded by the $x$-axis, the line $y=x$ and the circle $x^{2}+y^{2}=1.$

    MATH

  2. Find the volume of the solid bounded by the paraboloid $z=4-x^{2}-y^{2} $ and the $xy$-plane.

    MATH

  3. Find the volume inside the sphere MATH and outside the cylinder $x^{2}+y^{2}=9.$

    MATH

  4. Find the volume inside the sphere MATH and outside the cylinder MATH

    [This is a project problem but a hint is to write the equation of the cylinder in polar coordinates.]

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