Multivariable Calculus 15.1 Textbook Notes

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15.1 Double and Iterated Integrals over Rectangles

In chapter 15, we will be studying multiple integration. The first step in that process is to understand the double integral. In single-variable calculus, you learned that the integral of $y=f(x)$ represents the area between the graph of the function and the x-axis. If the area was above the axis, we counted it as positive area. If the area was below the axis, we b counted it as negative. Our definite integrals were written as a ${\int }_a^{b}f(x)dx$, a notation that meant that we were integrating the function with respect to x on the interval $[a,b]$. Really this came down to the area bounded by the lines $x=a,x=b,\text{ and }y=0$ and the curve $y=f(x)$ with positive/negative considerations discussed above. Now, our functions are of the form $z=f(x,y)$ and their graphs live in ${\mathbb{R}}^3$. When we integrate a function, we will need to look at a domain which is a subset of ${\mathbb{R}}^2$, which we understand to be the $xy$-coordinate plane.

The definition is done in a similar way as in the one-variable case: partition the region $\mathcal{R}$ into the non-overlapping union of small rectangles, form the Riemann sum of the integrand $f(x,y)$, and take the limit when the partition size goes to $0$. If the Riemann sum has a limit, that limit is called the integral of $f$ over the region $\mathcal{R}$, and is denoted as ${\iint }_{\mathcal{R}}f(x,y)dA$ or ${\iint }_{\mathcal{R}}f(x,y)dxdy$.

Also as in one-variable calculus, one uses the definition to relate the integral to the volume under the graph of $f$ over $\mathcal{R}$ and establishes integrability criteria and properties of the integral, but rarely evaluates an integral through this process. Instead, one develops a method of evaluating iterated single variable integrals in the form of ${\int }_a^b{\int }_c^{d}f(x,y)dydx\text{ if }\mathcal{R}=[a,b]\times [c,d]$ to evaluate ${\iint }_{\mathcal{R}}f(x,y)dA$. This is called Fubini’s Theorem. Here, in ${\int }_c^{d}f(x,y)dy$, we hold $x$ fixed and integrate out $f(x,y)$ as a function of $y$, then treat the outcome as a function of x and integrate it over x.

Find the following definitions/concepts/formulas/theorems:

• partition (you should have seen this when you defined the integral in Calculus I
• Riemann sum (same general idea as single-variable, but our little intervals are now little rectangles)
• limit of Riemann sums
• norm (of a partition)
• integrable (essentially the same as single-variable)
• double integral
• area element (dA)
• Continuous functions are integrable
• iterated integral
• Theorem: Fubini’s Theorem
• Double integral over a rectangle

The first few pages are all about how we extend our understanding of integration and Riemann sums to functions of two variables. There is a lot of notation, but it is really close enough to the single-variable case that it should be readable. The two examples are very basic, and you should not have any trouble working through them. Notice that integration of a multivariable expression with respect to one of its variables treats the other one as a constant. This is for all of the same reasons as why we take partial derivatives the way that we do, and it should not come as a great shock to you.

Reading Questions/Quizzes

1. Evaluate ${\int }_0^1{x}^{2}ydy$ and then ${\int }_0^2{\int }_0^1{x}^{2}ydydx$ as well as ${\int }_0^1{\int }_0^2{x}^{2}ydxdy$.
2. Evaluate ${\int }_0^1{\int }_0^{\pi }x\sin (xy)dydx$ and then ${\int }_0^{\pi }{\int }_0^{1}x\sin (xy)dxdy$
3. Evaluate ${\int }_0^1{\int }_0^1\frac{y}{1+xy}dydx$ and then ${\int }_0^1{\int }_0^1\frac{y}{1+xy}dxdy$