Table of Contents

Evaluating Double Integrals over Rectangles Examples 1
Recall from the Fubini’s Theorem and Evaluating Double Integrals over Rectangles page that if $z = f(x, y)$ is a two variable realvalued function and $f$ is continuous on the rectangle $R = [a, b] \times [c, d]$, then:
(1)
Furthermore, if $f$ can be written as a product of the single variable functions $g(x)$ and $h(y)$, that is $f(x, y) = g(x) h(y)$, then:
(2)
We will now look at some examples of evaluating double integrals over rectangles. More examples can be found on the Evaluating Double Integrals over Rectangles Examples 2 page.
Example 1
Evaluate the double integral $\iint_R \frac{xe^x}{y} \: dy \: dx$ where $R = [0, 1] \times [1, 2]$.
We can set the double integral above as an iterated integral as follows:
(3)
Let’s first evaluate the inner integral $\int_1^2 \frac{xe^x}{y} \: dy$ holding $x$ as fixed
(4)
Therefore we have that:
(5)
Note that $\int xe^x \: dx = xe^x – e^x$ can be obtained by using either Integration by Parts or Tabular Integration.
Example 2
Evaluate the double integral $\iint_R \cos (x – y) \: dA$ where $R = \left [ 0 , \frac{\pi}{2} \right ] \times \left [0, \frac{\pi}{2} \right ]$.
We can rewrite this double integral as an iterated integral:
(6)
We will first evaluate the inner integral $\int_0^{\frac{\pi}{2}} \cos (x – y) \: dx$ by holding $y$ as fixed as follows:
(7)
Therefore we have that:
(8)
Example 3
Evaluate the double integral $\iint_R x^2 y^3 \: dA$ where $R = \left [ 1, 2 \right ] \times \left [1,3 \right ]$.
This integral can be rewritten and evaluated as:
(9)
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