# Iterated Integrals

We are about to look at a method of evaluating double integrals over rectangles and more general domains without using the definition of a double integral, but before we do so, we will first need to learn what an iterated integral.

Suppose that $z = f(x, y)$ is a two variable real-valued function, and suppose that $f$ is integrable for $a ≤ x ≤ b$ and $c ≤ y ≤ d$, that is $f$ is integrable over the rectangle:

(1)

Just like partial differentiation with respect to a specific variable, we can also partial integrate with respect to a specific variable. For example, $\int_c^d f(x, y) \: dy$ means that we integrate $f$ with respect to $y$ while holding the variable $x$ as fixed. When we evaluate this integral, we will obtain a function in terms of $x$ only, and hence, we could then integrate the result from $a$ to $b$ with respect to $x$ as:

(2)

The result above is what we call an iterated integral which we will define below.

Definition: If $z = f(x, y)$ is a two variable real valued function, and if $f$ is integrable over the rectangle $R = [a, b] \times [c, d]$, then the Iterated Integral of $f$ over $R$ is $\int_a^b \int_c^d f(x, y) \: dy \: dx$. |

*We will see later that iterated integrals need not be over rectangles but instead can be done over more general domains.*

One important property about iterated integrals is that we can partially integrate $f(x, y)$ with respect to either variable $x$ or $y$ first, and then continue onward with integrating with respect to the second variable, that is:

(3)

However, it is important to note that **sometimes partial integrating with respect to a certain variable first will be a much easier process**.

Let’s now look at some examples of evaluating iterated integrals.

## Example 1

**Evaluate the following iterated integral $\int_2^4 \int_1^3 x^3 + xy^2 \: dy \: dx$. Over what rectangle is $f(x, y) = x^3 + xy^2$ being integrated?**

When evaluated iterated integrals over rectangles, we always want to work from the inside out. Let’s first evaluate the inside integral with respect to $y$ while holding $x$ as fixed.

(4)

Therefore we have that:

(5)

Evaluating this definite integral and we get that:

(6)

Therefore we have that:

(7)

In this particular example, we are integrating over the rectangle $R = [2, 4] \times [1, 3]$.

## Example 2

**Evaluate the following iterated integral $\int_0^{\pi} \int_{0}^{\pi} \sin x \cos y \: dx \: dy$. Over what rectangle is $f(x, y) = \sin x \cos y$ being integrated?**

We will first start by evaluating the inner integral while holding $y$ as fixed:

(8)

Therefore we have that:

(9)

Evaluating this definite integral and we get that:

(10)

In this particular example, we are integrating over the square $R = [0, \pi] \times [0, \pi]$.