Iterated Integrals over Rectangles

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Notation and Order

Let's review our notation for the different kinds of integrals:

$\displaystyle{\iint_R f(x,y) \, dA}$ is the double integral of $f(x,y)$ over the region $R$.

$\displaystyle{\iint_R f(x,y) \, dx\, dy}$ and $\displaystyle{\iint_R f(x,y) \, dy \, dx}$ mean the exact same thing as $\displaystyle{\iint_R f(x,y) \, dA}$. The area of a little box is $\Delta A = \Delta x \,\Delta y = \Delta y\, \Delta x$, so the infinitesimal area is $dA = dx\, dy = dy\, dx$.

$\displaystyle{\int_a^b \int_c^d f(x,y)\, dy \, dx}$ is an iterated integral. We first treat $x$ as a constant and integrate $f(x,y) \,dy$ from $y=c$ to $y=d$. Call the result $g(x)$. It is a function of $x$ and describes how much stuff is in the column with that value of $x$. We then integrate $g(x)\, dx$ from $x=a$ to $x=b$.

$\displaystyle{\int_c^d \int_a^b f(x,y)\, dx \, dy}$ is an iterated integral where we first integrate over $x$ to get the contribution of a row, and then integrate over $y$ to add up all the rows.

In an iterated integral, the order of $dx$ and $dy$ tells you which variable to integrate first. When in doubt, draw parentheses: $$\int_a^b \int_c^d f(x,y) \, dy\, dx = \int_a^b \left ( \int_c^d f(x,y) \,dy \right )\, dx.$$

Fubini's Theorem says that you can evaluate double integrals by doing an iterated integral in either order, but sometimes one order is a lot simpler than the other.

In the following video, we go over notation (2 minutes) and work two example problems (4 minutes each). The first example involves integrating $6x^2y$ over the rectangle $[1,3] \times [0,2]$, and both orders of integration are equally easy. The second example involves integrating $x e^{xy}$ over the same rectangle, and one order turns out to be much easier than the other.

The Fundamental Theorem of Calculus

The Indefinite Integral and the Net Change

Substitution

Area Between Curves

Volumes

Integration by Parts

Integrals of Trig Functions

Trig Substitutions

Partial Fractions

Strategies of Integration

Improper Integrals

Differential Equations

Models of Growth

Infinite Sequences

Infinite Series

Integral Test

Comparison Tests

Convergence of Series with Negative Terms

The Ratio and Root Tests

Strategies for testing Series

Power Series

Representing Functions as Power Series

Taylor and Maclaurin Series

Applications of Taylor Polynomials

Partial Derivatives

Multiple Integrals

Iterated Integrals over Rectangles

Double Integrals over General Regions

Notation and Order