1.1Evaluate a Rectangle Integral
Evaluate $\int_0^1\int_0^2 (x+y)\,dy\,dx$.
Solution
Integrating in $y$ gives $2x+2$. Integrating from $0$ to $1$ gives $3$.
Multiple Integrals Practice
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Difficulty
Direct Practice
1.2Compute an Average Value
Find the average value of $f(x,y)=x+y$ on the unit square.
Solution
$\iint_{[0,1]^2}(x+y)\,dA=1$, and the area is $1$, so the average is $1$.
Difficulty
Integrated Practice
2.1Use a Polar Jacobian
Write the polar-coordinate integral for the area of the disk $x^2+y^2\le4$.
Solution
The bounds are $0\le r\le2$ and $0\le\theta\le2\pi$, so the area is $\int_0^{2\pi}\int_0^2 r\,dr\,d\theta$.
2.2Set Up a Mass Integral
Set up the mass of the rectangle $0\le x\le2$, $0\le y\le1$ with surface density $\rho(x,y)=x+y$.
Solution
$m=\int_0^2\int_0^1(x+y)\,dy\,dx$.
Difficulty
Applied Problems
3.1Choose Spherical Bounds
Set up the volume of the sphere $x^2+y^2+z^2\le a^2$ using the convention in the note.
Solution
$\int_0^{2\pi}\int_0^\pi\int_0^a \rho^2\sin\phi\,d\rho\,d\phi\,d\theta$.
Difficulty
Challenge / Synthesis
4.1Reverse an Iterated Integral
Reverse the order of integration for $\int_0^1\int_x^1 f(x,y)\,dy\,dx$.
Solution
The region is $0\le x\le y\le1$, so the reversed integral is $\int_0^1\int_0^y f(x,y)\,dx\,dy$.