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Linear Algebra for MATH 308
Lecture 1: Vectors, Linear Independence, and Spanning Sets
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Section 10: Moment Generating Function
Section 11: Markov’s Inequality, Chebyshev’s Inequality, and Weak Law of Large Numbers
Section 12: Convergence and the Central Limit Theorem
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Double Integrals
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
.
Concepts
Two Dimensional Integrals Over Rectangles
Two Dimensional Integrals Over Other Regions
Changing Order of Integration
Two Dimensional Integrals – Change of Variables
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.
Compute \(\displaystyle \int_{y=2}^{3}\int_{x=1}^{2}x^2y\,dxdy.\)
Compute \(\displaystyle \int_{x=1}^{2}\int_{y=2}^{3}x^2y\,dydx\)
Reveal Answers
\(\dfrac{35}{6}\)
\(\dfrac{35}{6}\)
Watch Video
To see the full video page and find related videos, click the following link.
Mechanics of Computing Iterated Integrals
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 \([1,1.5]\times[2.5,3]?\)
Reveal Answers
\(\dfrac{35}{6}\) lb.
\(\dfrac{209}{192}\) lb.
Watch Video
To see the full video page and find related videos, click the following link.
Using Double Integrals to Compute Mass and Weight
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 \(\dfrac{1}{3}<Y<1?\)
Reveal Answers
See the video for an explanation.
\(\dfrac{3}{8}\)
Watch Video
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.
Compute \(\displaystyle \int_{x=0}^{1}\int_{y=x^2}^{x}x^2 y\, dydx.\)
Compute \(\displaystyle \int_{y=0}^{1}\int_{x=y}^{\sqrt{y}}x^2y\,dxdy.\)
Reveal Answers
\(\dfrac{1}{35}\)
\(\dfrac{1}{35}\)
Watch Video
To see the full video page and find related videos, click the following link.
Changing Order of Integration for Iterated Integrals
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 \(y=x?\)
Reveal Answer
\(\dfrac{5}{8}\) lb.
Watch Video
To see the full video page and find related videos, click the following link.
Determining the Iterated Integral for a Given Region
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?
Reveal Answers
\(1-\dfrac{2}{e}\)
\(1-\dfrac{1+z}{e^z}\). As \(z\rightarrow \infty\), the value of the integral approaches 1.
Watch Video
To see the full video page and find related videos, click the following link.
Using a Parameter to Compute an Improper Double Integral
Let \(f(x,y) = x+y\) on \([0,2]\times[0,2].\) Find the integral of \(f\) over the region \(xy \leq 1.\)
Reveal Answer
\(3.5\)
Watch Video
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.
What is Fubini's Theorem?
Compute \(\displaystyle \int_{x=0}^{1}\int_{y=x}^{1}e^{y^2}\,dydx\)
Reveal Answer
See the video for an explanation.
\(\dfrac{1}{2}[e-1]\)
Watch Video
To see the full video page and find related videos, click the following link.
Fubini's Theorem and Changing Order of Integration
Find \(\displaystyle \int_{x=1/2}^{2}\int_{y=1/x}^{2}y\cos(xy)\,dydx.\)
Reveal Answer
\(\dfrac{1}{2}[\cos(1)-\cos(4)]-\dfrac{3}{2} \sin (1)\)
Watch Video
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.
Find the integral of \(f(x,y) = x^2 + y^2\) over the disk of radius \(1\) centered at the origin.
Reveal Answer
\(\dfrac{\pi}{2}\)
Watch Video
To see the full video page and find related videos, click the following link.
Double Integral Using Polar Coordinates
Find the integral of \(f(x,y)=xy\) in the disk of radius \(1\) centered at the origin and in the first quadrant.
Reveal Answer
\(\dfrac{1}{8}\)
Watch Video
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.
\(\mathcal{R}\) 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.
Reveal Answer
See the video for the region and Jacobian.
Watch Video
To see the full video page and find related videos, click the following link.
Change of Variables for Double Integrals
\(\mathcal{R}\) 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 \(y=\dfrac{1}{2}(u-v).\)
Reveal Answer
See the video for the region and the Jacobian.
Watch Video
To see the full video page and find related videos, click the following link.
Change of Variables for Double Integrals
What is the change of variables formula in two variables?
Reveal Answer
See the video for an explanation.
Watch Video
To see the full video page and find related videos, click the following link.
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 \(y = 2u-3v.\)
Reveal Solution
\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*}
Watch Video
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.