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MULTIPLE INTEGRALS 2.2 Iterated Integrals In this section, we will learn how to: Express double integrals as iterated integrals.
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INTRODUCTION Once we have expressed a double integral as an iterated integral, we can then evaluate it by calculating two single integrals.
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INTRODUCTION Suppose that f is a function of two variables that is integrable on the rectangle R = [a, b] x [c, d]
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INTRODUCTION We use the notation to mean: x is held fixed f(x, y) is integrated with respect to y from y = c to y = d
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PARTIAL INTEGRATION This procedure is called partial integration with respect to y. Notice its similarity to partial differentiation.
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PARTIAL INTEGRATION Now, is a number that depends on the value of x. So, it defines a function of x:
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PARTIAL INTEGRATION If we now integrate the function A with respect to x from x = a to x = b, we get: Equation 1
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ITERATED INTEGRAL The integral on the right side of Equation 1 is called an iterated integral.
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ITERATED INTEGRALS Thus, means that: First, we integrate with respect to y from c to d. Then, we integrate with respect to x from a to b. Equation 2
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ITERATED INTEGRALS Similarly, the iterated integral means that: First, we integrate with respect to x (holding y fixed) from x = a to x = b. Then, we integrate the resulting function of y with respect to y from y = c to y = d.
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ITERATED INTEGRALS Example 1
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FUBUNI’S THEOREM If f is continuous on the rectangle R = {(x, y) |a ≤ x ≤ b, c ≤ y ≤ d} then Theorem 4
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ITERATED INTEGRALS Example 2
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ITERATED INTEGRALS Example 3
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ITERATED INTEGRALS To be specific, suppose that: f(x, y) = g(x)h(y) R = [a, b] x [c, d]
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ITERATED INTEGRALS Then, Fubini’s Theorem gives:
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ITERATED INTEGRALS In the inner integral, y is a constant. So, h(y) is a constant and we can write: since is a constant.
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ITERATED INTEGRALS Hence, in this case, the double integral of f can be written as the product of two single integrals: where R = [a, b] x [c, d] Equation 5
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ITERATED INTEGRALS Example 4
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