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 and watch the video for an explanation.
- Compute \(\displaystyle \int_{y=2}^{3}\int_{x=1}^{2}x^2y\,dxdy.\)
- Compute \(\displaystyle \int_{x=1}^{2}\int_{y=2}^{3}x^2y\,dydx\)
- 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?
- What is the weight of the portion of this plate that lies on \([1,1.5]\times[2.5,3]?\)
- 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.
- What is the probability of the event \(0<X<\dfrac{3}{4}\) and \(\dfrac{1}{3}<Y<1?\)
Two Dimensional Integrals Over Other Regions
Examples
Directions: You should attempt to solve the problems first, and then you can check your solution and watch the video for an explanation.
- Compute \(\displaystyle \int_{x=0}^{1}\int_{y=x^2}^{x}x^2 y\, dydx.\)
- Compute \(\displaystyle \int_{y=0}^{1}\int_{x=y}^{\sqrt{y}}x^2y\,dxdy.\)
- 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 \(y=x?\)
- Let \(f(x,y) = e^{-x-y}\) on \([0,\infty)\times[0,\infty).\) What is the integral of \(f\) over the region \(x+y\leq 1?\)
- 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?
- Let \(f(x,y) = x+y\) on \([0,2]\times[0,2].\) Find the integral of \(f\) over the region \(xy \leq 1.\)
Changing Order of Integration
Examples
Directions: You should attempt to solve the problems first, and then you can check your solution and watch the video for an explanation.
- What is Fubini's Theorem?
- Compute \(\displaystyle \int_{x=0}^{1}\int_{y=x}^{1}e^{y^2}\,dydx\)
- Find \(\displaystyle \int_{x=1/2}^{2}\int_{y=1/x}^{2}y\cos(xy)\,dydx.\)
Two Dimensional Integrals – Change of Variables
Examples
Directions: You should attempt to solve the problems first, and then you can check your solution and watch the video for an explanation.
- Find the integral of \(f(x,y) = x^2 + y^2\) over the disk of radius \(1\) centered at the origin.
- Find the integral of \(f(x,y)=xy\) in the disk of radius \(1\) centered at the origin and in the first quadrant.
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.
- \(\mathcal{R}\) 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.
- \(\mathcal{R}\) 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 \(y=\dfrac{1}{2}(u-v).\)
- What is the change of variables formula in two variables?
- 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 \(y = 2u-3v.\)