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Chapter 15 – Multiple Integrals

Chapter 15 – Multiple Integrals. 15.7 Triple Integrals. Objectives: Understand how to calculate triple integrals Understand and apply the use of triple integrals to different applications. Triple Integrals.

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Chapter 15 – Multiple Integrals

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  1. Chapter 15 – Multiple Integrals 15.7 Triple Integrals • Objectives: • Understand how to calculate triple integrals • Understand and apply the use of triple integrals to different applications 15.7 Triple Integrals

  2. Triple Integrals • Just as we defined single integrals for functions of one variable and double integrals for functions of two variables, so we can define triple integrals for functions of three variables. 15.7 Triple Integrals

  3. Triple Integrals • Let’s first deal with the simplest case where f is defined on a rectangular box: 15.7 Triple Integrals

  4. Triple Integrals • The first step is to divide Binto sub-boxes—by dividing: • The interval [a, b] into lsubintervals [xi-1, xi] of equal width Δx. • [c, d] into msubintervals of width Δy. • [r, s] into nsubintervals of width Δz. 15.7 Triple Integrals

  5. Triple Integrals • The planes through the endpoints of these subintervals parallel to the coordinate planes divide the box B into lmn sub-boxes • Each sub-box has volume ΔV = ΔxΔyΔz 15.7 Triple Integrals

  6. Triple Integrals • Then, we form the triple Riemann sum where the sample point is in Bijk. 15.7 Triple Integrals

  7. Triple Integrals • The triple integralof f over the box B is: if this limit exists. • Again, the triple integral always exists if fis continuous. 15.7 Triple Integrals

  8. Fubini’s Theorem for Triple Integrals • Just as for double integrals, the practical method for evaluating triple integrals is to express them as iterated integrals, as follows. • If f is continuous on the rectangular box B = [a, b] x [c, d] x [r, s], then 15.7 Triple Integrals

  9. Fubini’s Theorem • The iterated integral on the right side of Fubini’s Theorem means that we integrate in the following order: • With respect to x (keeping y and z fixed) • With respect to y (keeping z fixed) • With respect to z 15.7 Triple Integrals

  10. Example 1 – pg. 998 # 4 • Evaluate the triple integral. 15.7 Triple Integrals

  11. Integral over a Bounded Region • We restrict our attention to: • Continuous functions f • Certain simple types of regions 15.7 Triple Integrals

  12. Type I Region Eq.5 • A solid region Eis said to be of type 1if it lies between the graphs of two continuous functions of x and y. That is, where D is the projection of E onto the xy-plane. 15.7 Triple Integrals

  13. Type I Region • Notice that: • The upper boundary of the solid E is the surface with equation z = u2(x, y). • The lower boundary is the surface z = u1(x, y). 15.7 Triple Integrals

  14. Type I Region Eq. 6 • If E is a type 1 region given by Equation 5, then we have Equation 6: 15.7 Triple Integrals

  15. Type II Region • A solid region Eis said to be of type 1if it lies between the graphs of two continuous functions of x and y. That is, where D is the projection of E onto the yz-plane. 15.7 Triple Integrals

  16. Type II Region • Notice that: • The back surface is x = u1(y, z). • The front surface is x = u2(y, z). 15.7 Triple Integrals

  17. Type II Region Eq. 10 • For this type of region we have: 15.7 Triple Integrals

  18. Type III Region • Finally, a type 3region is of the form where: • D is the projection of Eonto the xz-plane. 15.7 Triple Integrals

  19. Type III Region • Notice that: • y = u1(x, z) is the left surface. • y = u2(x, z) is the right surface. 15.7 Triple Integrals

  20. Type III Region Eq. 11 • For this type of region, we have: 15.7 Triple Integrals

  21. Visualization • Regions of Integration in Triple Integrals 15.7 Triple Integrals

  22. Example 2 – pg. 998 # 10 • Evaluate the triple integral. 15.7 Triple Integrals

  23. Example 3 – pg. 998 # 20 • Use a triple integral to find the volume of the given solid. 15.7 Triple Integrals

  24. Example 4 – pg. 998 # 22 • Use a triple integral to find the volume of the given solid. 15.7 Triple Integrals

  25. Example 5 – pg. 999 # 36 • Write five other iterated integrals that are equal to the given iterated integral. 15.7 Triple Integrals

  26. More Examples The video examples below are from section 15.7 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. • Example 1 • Example 3 • Example 6 15.7 Triple Integrals

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