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Chapter 15 – Multiple Integrals. 15.2 Iterated Integrals. Objectives: Express double integrals as Iterated integrals. Partial Integration. If we now integrate the function A with respect to x from x = a to x = b , we get:
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Chapter 15 – Multiple Integrals 15.2 Iterated Integrals • Objectives: • Express double integrals as Iterated integrals 15.2 Iterated Integrals
Partial Integration • If we now integrate the function A with respect to x from x = a to x = b, we get: • The integral on the right side is called an iterated integral. 15.2 Iterated Integrals
Iterated Integrals • Usually, the brackets are omitted. • Thus, • means that: • First, we integrate with respect to y from c to d. • Then, we integratewith respect to x from a to b. 15.2 Iterated Integrals
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, weintegrate the resulting function of ywith respect to y from y = c to y = d. 15.2 Iterated Integrals
Fubini’s Theorem • Visualization • Fubini’s Theorem 15.2 Iterated Integrals
Equation 5 • Double integral of f can be written as the product of two single integrals since is a constant. 15.2 Iterated Integrals
Example 1 – pg 987 • Calculate the iterated integral. • #4. • #6. 15.2 Iterated Integrals
Example 2 – pg. 987 • Calculate the double integral. • . • #20. 15.2 Iterated Integrals
Example 3 • Find the volume of the solid that lies under the hyperbolic paraboloid z = 4 + x2 - y2and above the rectangle R=[-1,1] x [0,2] 15.2 Iterated Integrals
Example 4 – pg. 988 # 36 • Find the average value of f over the given rectangle. 15.2 Iterated Integrals
More Examples The video examples below are from section 14.6 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. • Example 2 • Example 3 • Example 4 15.2 Iterated Integrals