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Chapter 15 – Multiple Integrals

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

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  1. Chapter 15 – Multiple Integrals 15.2 Iterated Integrals • Objectives: • Express double integrals as Iterated integrals 15.2 Iterated Integrals

  2. 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

  3. 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

  4. 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

  5. Fubini’s Theorem • Visualization • Fubini’s Theorem 15.2 Iterated Integrals

  6. Equation 5 • Double integral of f can be written as the product of two single integrals since is a constant. 15.2 Iterated Integrals

  7. Example 1 – pg 987 • Calculate the iterated integral. • #4. • #6. 15.2 Iterated Integrals

  8. Example 2 – pg. 987 • Calculate the double integral. • . • #20. 15.2 Iterated Integrals

  9. 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

  10. Example 4 – pg. 988 # 36 • Find the average value of f over the given rectangle. 15.2 Iterated Integrals

  11. 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

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