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Double Integrals

Instructions

  • This workshop is a brief review or introduction to double integrals.
  • You can attempt to solve the problems first, and then check your answers. Each example also has a video explaining the problem. 
  • If you want to see more videos on this topic you can see the material in Sections 15.1–15.9 of our Math 251 course. 
Two Dimensional Integrals Over Rectangles

Examples


Directions: You should attempt to solve the problems first, and then you can check your solution. You can also watch the video for an explanation of the problems.
  1. Compute \(\displaystyle \int_{y=2}^{3}\int_{x=1}^{2}x^2y\,dxdy.\)
  2. Compute \(\displaystyle \int_{x=1}^{2}\int_{y=2}^{3}x^2y\,dydx\)

    1. \(\dfrac{35}{6}\)
    2. \(\dfrac{35}{6}\)

    To see the full video page and find related videos, click the following link.
    Mechanics of Computing Iterated Integrals

  3. Assume a plate lies on the rectangle \([1,2]\times[2,3]\) and has weight density \(yx^2 \frac{\textnormal{pounds}}{\textnormal{foot}^2}\) What is the weight of this plate?
  4. What is the weight of the portion of this plate that lies on \([1,1.5]\times[2.5,3]?\)

    1. \(\dfrac{35}{6}\) lb.
    2. \(\dfrac{209}{192}\) lb.


    To see the full video page and find related videos, click the following link.
    Using Double Integrals to Compute Mass and Weight

  5. Let \(f(x,y) = 6x^2 y\) on \([0,1]\times[0,1]\) and \(0\) everywhere else, be the p.d.f. of the two random variables \(X\) and \(Y.\) Verify that \(f\) is a p.d.f.
  6. What is the probability of the event \(0<X<\dfrac{3}{4}\) and \(\dfrac{1}{3}<Y<1?\)

    1. See the video for an explanation.
    2. \(\dfrac{3}{8}\)


    To see the full video page and find related videos, click the following link.
    Applications of Double Integrals to Probability

Two Dimensional Integrals Over Other Regions

Examples


Directions: You should attempt to solve the problems first, and then you can check your solution. You can also watch the video for an explanation of the problems.
  1. Compute \(\displaystyle \int_{x=0}^{1}\int_{y=x^2}^{x}x^2 y\, dydx.\)
  2. Compute \(\displaystyle \int_{y=0}^{1}\int_{x=y}^{\sqrt{y}}x^2y\,dxdy.\)

    1. \(\dfrac{1}{35}\)
    2. \(\dfrac{1}{35}\)


    To see the full video page and find related videos, click the following link.
    Changing Order of Integration for Iterated Integrals

  3. Assume a plate that lies on the square \([0,1]\times[0,1]\) has weight density function \(1 + yx \frac{\textnormal{pounds}}{\textnormal{feet}^2}.\) What is the weight of the portion of this plate that lies above the line \(y=x?\)

    \(\dfrac{5}{8}\) lb.


    To see the full video page and find related videos, click the following link.
    Determining the Iterated Integral for a Given Region

  4. Let \(f(x,y) = e^{-x-y}\) on \([0,\infty)\times[0,\infty).\) What is the integral of \(f\) over the region \(x+y\leq 1?\)
  5. Let \(f(x,y) = e^{-x-y}\) on \([0,\infty)\times[0,\infty).\) What is the integral of \(f\) over the region \(x+y\leq z\), where \(z\) is any non–negative number?

    1. \(1-\dfrac{2}{e}\)
    2. \(1-\dfrac{1+z}{e^z}\). As \(z\rightarrow \infty\), the value of the integral approaches 1.


    To see the full video page and find related videos, click the following link.
    Using a Parameter to Compute an Improper Double Integral

  6. Let \(f(x,y) = x+y\) on \([0,2]\times[0,2].\) Find the integral of \(f\) over the region \(xy \leq 1.\)

    \(3.5\)


    To see the full video page and find related videos, click the following link.
    Splitting a Domain Into Two Pieces for a Double Integral

Changing Order of Integration

Examples


Directions: You should attempt to solve the problems first, and then you can check your solution. You can also watch the video for an explanation of the problems.
  1.  What is Fubini's Theorem?
  2. Compute \(\displaystyle \int_{x=0}^{1}\int_{y=x}^{1}e^{y^2}\,dydx\)

    1. See the video for an explanation.
    2. \(\dfrac{1}{2}[e-1]\)


    To see the full video page and find related videos, click the following link.
    Fubini's Theorem and Changing Order of Integration

  3. Find \(\displaystyle \int_{x=1/2}^{2}\int_{y=1/x}^{2}y\cos(xy)\,dydx.\)

    \(\dfrac{1}{2}[\cos(1)-\cos(4)]-\dfrac{3}{2} \sin (1)\)


    To see the full video page and find related videos, click the following link.
    Changing Order of Integration for an Iterated Integral

Two Dimensional Integrals – Change of Variables

Examples


Directions: You should attempt to solve the problems first, and then you can check your solution. You can also watch the video for an explanation of the problems.
  1. Find the integral of \(f(x,y) = x^2 + y^2\) over the disk of radius \(1\) centered at the origin. 

    \(\dfrac{\pi}{2}\)


    To see the full video page and find related videos, click the following link.
    Double Integral Using Polar Coordinates

  2. Find the integral of \(f(x,y)=xy\) in the disk of radius \(1\) centered at the origin and in the first quadrant.

    \(\dfrac{1}{8}\)


    To see the full video page and find related videos, click the following link.
    Double Integral Using Polar Coordinates

In examples 3 and 4, determine the new region that we get by applying the given transformation to the region \(\mathcal{R}.\) Also, find the Jacobian of the transformation.
  1. \(\mathcal{R}\) is the region \(x^2 + \dfrac{y^2}{36}=1\) and the transformation is \(x=\dfrac{u}{2}\) and \(y=3v.\) Note the video also includes an explanation of the u-sub rule from Calculus 1.

    See the video for the region and Jacobian.


    To see the full video page and find related videos, click the following link.
    Change of Variables for Double Integrals

  2. \(\mathcal{R}\) is the region bounded by \(y=-x+4\), \(y=x+1\), and \(y=\dfrac{x}{3} - \dfrac{4}{3}\) and the transformation is \(x=\dfrac{1}{2}(u+v)\) and \(y=\dfrac{1}{2}(u-v).\)

    See the video for the region and the Jacobian.


    To see the full video page and find related videos, click the following link.
    Change of Variables for Double Integrals

  3. What is the change of variables formula in two variables?

    See the video for an explanation.


    To see the full video page and find related videos, click the following link.
    Change of Variables Formula in Two Variables

  4. Evaluate \(\iint\limits_{R}(x+y)dA\) where \(R\) is the trapezoidal region with vertices given by \((0,0),\) \((5/2,5/2),\) \((5/2,-5/2)\) and \((5,0)\) using the transformation \(x=2u+3v\) and \(y = 2u-3v.\)

    \begin{align*}
    \iint\limits_R x+y \,dA &=\int_{u=0}^{5/4} \int_{v=0}^{5/6} \left[(2u+3v)+(2u-3v)\right] \cdot 12\, dvdu\\[8pt]
    &=\int_{u=0}^{5/4} \int_{v=0}^{5/6} 48u \, dvdu\\[8pt]
    &=\dfrac{125}{4}
    \end{align*}
    Note:  In the video, the blue line (and upper limit of the integration for \(v\)) should be \(\frac{5}{6}\), not \(\frac{5}{3}\). Also, the final integral should be multiplied by the Jacobian, which is \(12\). You can see the Solution above for the correct equations.