Double Iterated Integrals Rectangular Region (Ex2)

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Examples of Double Iterated Integrals over Rectangular Regions (Example 2)

These kinds of double integrals have numbers as the integral boundaries for the x and y variables.

They are called iterated integrals, or sometimes integrals over a rectangular region. This is because the numbers in the integral bounds are each lines, which form a rectangle boundary when put together.

Example problem: Calculate the iterated integral: \(\int_1^3 {\int_1^5 {\frac{{\ln (y)}}{{xy}}} } dydx\)

Let’s start with the inside integral.

\[\int_1^5 {\frac{{\ln (y)}}{{xy}}dy} \]

So we integrate this with respect to y.

We can just use regular substitution.

Refresh yourself on the derivative for ln, the natural logarithm.

Remember to make new integral limits when performing substitution.

\[\begin{array}{l}u = \ln (y)\\du = \frac{1}{y}dy\\u(1) = \ln (1) = 0\\u(4) = \ln (5)\end{array}\]

Substituting and integrating:

\[\begin{array}{l}\frac{1}{x}\int_0^{\ln (5)} {udu} \\\frac{1}{x}\left[ {\frac{{{{\left[ {\ln (5)} \right]}^2}}}{2} – \frac{{{{(0)}^2}}}{2}} \right]\\\frac{{{{\left[ {\ln (5)} \right]}^2}}}{{2x}}\\I = \int_1^3 {\frac{{{{\left[ {\ln (5)} \right]}^2}}}{{2x}}dx} \end{array}\]

That’s the inner integral, and we just plug it back into the original.

Let’s call the original integral “I”.

We can take the constant terms out of the integral as well.

\[\begin{array}{l}I = \frac{{{{\left[ {\ln (5)} \right]}^2}}}{2}\int_1^3 {\frac{1}{x}} dx\\I = \frac{{{{\left[ {\ln (5)} \right]}^2}}}{2} \cdot \left[ {\ln (3) – \ln (1)} \right]\\I = \frac{{{{\left[ {\ln (5)} \right]}^2}\ln (3)}}{2}\end{array}\]

And we’ve solved the integral.

You can’t simplify these logarithms any further, but it would be handy to look for logarithm laws and memorize how to simplify them.

Try an example where you find a double integral over a different region, such as an ellipse region!

Try using integrals to find the volumes between a cylinder and ellipsoid!

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