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Examples of Double Iterated Integrals over Rectangular Regions (Example 1)
These kinds of double integrals (“iterated” integrals) have numbers as the integral boundaries for the x and y variables and aren’t difficult, but end up having a lot of arithmetic that’s easy to make a mistake with.
You know the region of integration is a “rectangle” because all four limits of integration are just numbers (the numbers are just 4 lines that form a boundary, which is a rectangle).
Example Problem: Calculate the iterated integral \[\int_1^4 {\int_0^2 {(6{x^2}y – 2x)dydx} } \]
Here, the region of integration is a rectangle, formed by the boundaries x=1, x=4, y=0, and y=2.
Start off with the “inner” integral.
Pay attention to the order of the variables, here it’s dydx, so when doing the inside integral integrate with respect to the “y” variable. You may be given problems where it’s dxdy at the end, watch out!
So we start off with the “inside” integral, by doing:
\[\int_0^2 {(6{x^2}y – 2x)dy} \]
And you really just integrate with respect to the y variable, and if a term doesn’t have any y term then you multiply it by a y term.
\[\begin{array}{l}\int_1^4 {3{x^2}{{(2)}^2} – 2x(2) – 3{x^2}{{(0)}^2} + 2x(0)} dx\\\int_1^4 {(12{x^2} – 4x)} dx\end{array}\]
Now we’re left with a normal looking integral, which we continue to solve normally.
\[\begin{array}{l}4{(4)^3} – 2{(4)^2} – 4{(1)^3} + 2{(1)^2}\\ = 222\end{array}\]
Not so bad, but the first inner integral you do will be a bit of algebra so be wary of algebraic mistakes and be cautious of the order of the integral and check if it’s dxdy or dydx.
Try another double iterated integral over a rectangle region example!
Try an example where you find a double integral over a different region, such as an ellipse region!