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Gerald has taught engineering, math and science and has a doctorate in electrical engineering.
The word ''iteration'' means to do something repeatedly. This word is often used in math. In this lesson, we evaluate a double integral by using iteration. First, we'll do a single integral on one of the variables and then repeat with a single integral on the other variable. Rather than repeat this information, let's do an example.
Imagine needing to find the solution to the double integral
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We have two variables, x and y. Let's integrate on x first. To be clear, we can use parentheses:
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The integral inside the parentheses is called the inner integral. We need to evaluate the integral of yx2 from x = 2 to x = 3.
This is an integration on x. What about the y appearing in yx2? In this integration on x, we treat y as if it were a constant. So, yx2 is like cx2 where c is a constant.
The integration of cx2 is cx3/3. So, when we integrate on x, the integral of yx2 is yx3/3.
Evaluating with the upper and lower limits:
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Simplifying, we get 19y/3 as the answer for the inner integral. Next, we do the outer integral. We evaluate:
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Do you see how the integral within the parentheses has been replaced with 19y/3?
Now, we iterate (i.e, repeat) by integrating over y:
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Simplifying, we get 47.5 as the final answer.
The double integral was computed using an ''iterated integral.'' We started with an integration on x. What if we had started with an integration on y?
Then, the double integral would be written as:
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Do you see how the work is clarified by using parentheses and by labeling the lower limit of the integrals?
The integration on y becomes:
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Now, the variable is y while x is the constant. Continuing:
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Now, the outer integral is over x:
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Do you see how 15 divided by 3 produced 5?
Simplifying further, we get 47.5. Same as before. We can choose to iterate x then y or y then x. The result will be the same.
Something special happens when the function of x and y, f(x,y), can be written as the product of a function on x, g(x), multiplied with a function on y, h(y). Then, the iterative integral becomes the product of two integrals. This is really a special situation, but our example is just such a case. That's because we have f(x,y) = yx2 which is the same as g(x)h(y) where g(x) = x2 and h(y) = y. What we are saying is:
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Here are the details:
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Let's check:
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and
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Thus:
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Which is the same result as before.
Does this always happen? Yes, if the function being integrated can be written as a function of x multiplied by a function of y.
A double integral can be evaluated by repeated use of a single integral. This repetition is called an iterative integral. The variable integrated first can be written as an integral inside parentheses, called the inner integral. The remaining variable is integrated as the outer integral. When the function being integrated can be written as a function of x multiplied by a function of y, the iterated integral simplifies to the product of the integral of these separate functions. That is, if f(x,y) = g(x)h(y), then:
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GRE Math: Study Guide & Test Prep27 chapters | 182 lessons | 16 flashcard sets
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