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Chapter Twelve

Chapter Twelve. Multiple Integrals. Section 12.1 Double Integrals Over Rectangles. Goals Volumes and double integrals Midpoint Rule Average value Properties of double integrals. Volumes and Double Integrals. Given a function f (x, y), defined on a closed rectangle

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Chapter Twelve

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  1. Chapter Twelve Multiple Integrals

  2. Section 12.1Double Integrals Over Rectangles • Goals • Volumes and double integrals • Midpoint Rule • Average value • Properties of double integrals Calculus

  3. Volumes and Double Integrals • Given a function f (x, y), defined on a closed rectangle Suppose: f(x, y) ≥ 0. • Question: What is the volume of the solid S under the graph of f and above R? Calculus

  4. Volumes (cont’d) Calculus

  5. Volumes (cont’d) • We do this by • dividing the interval [a, b] into m subintervals [xi-1, xi] of equal width x = (b – a)/m and • dividing [c, d] into n subintervals [yj-1, yj] of equal width y = (d – c)/n. • Next we form the subrectangles each with area A = xy : Calculus

  6. Volumes (cont’d) Calculus

  7. Volumes (cont’d) • We choose a sample point • Then we can approximate the part of S that lies above each Rijby a thin rectangular box with base Rijand height • The volume of this box is the height of the box times the area of the base rectangle: Calculus

  8. Volumes (cont’d) Calculus

  9. Volumes (cont’d) • Following this procedure for all the rectangles and adding the volumes of the corresponding boxes, we get an approximation to the total volume of S: • This is illustrated on the next slide: Calculus

  10. Volumes (cont’d) Calculus

  11. Volumes (cont’d) • As m and n become larger and larger this approximation becomes better and better. • Thus we would expect that • We use this expression to define the volume of S. Calculus

  12. Double Integral • Limits of the preceding type occur frequently in a variety of settings, so we make the following general definition: Calculus

  13. Double Integral (cont’d) • A volume can be written as a double integral: Calculus

  14. Double Integral (cont’d) • The sum in our definition of double integral is called a double Riemann sum and is an approximation to the double integral. • If f happens to be a positive function, then the double Riemann sum is the sum of volumes of columns and approximates the volume under the graph of f. Calculus

  15. Example • Estimate the volume of the solid that lies • above the square R = [0, 2]  [0, 2] and • below the elliptic paraboloid z = 16 – x2 – 2y2. • Divide R into four equal squares and choose the sample point to be the upper right corner of each square Rij. • Sketch the solid and the approximating rectangular boxes. Calculus

  16. Solution • The squares are shown on the next slide. • The paraboloid is the graph off(x, y) = 16 – x2 – 2y2 and the area of each square is 1. Approximating the volume by the Riemann sum with m = n = 2, we have Calculus

  17. Solution (cont’d) Calculus

  18. Solution (cont’d) • Thus 34 is thevolume of theapproximatingrectangular boxesshown: Calculus

  19. Using More Squares • We get better approximations to the volume in the preceding example if we increase the number of squares. • The next slides show how the columns start to look more like the actual solid when we use 16, 64, and 256 squares: Calculus

  20. Using More Squares (cont’d) Calculus

  21. Using More Squares (cont’d) Calculus

  22. The Midpoint Rule • We use a double Riemann sum to approximate the double integral. • The sample pointto be the center Calculus

  23. Example • Use the Midpoint Rule with m = n = 2 to estimate the value of • Solution We evaluate f(x, y) = x – 3y2 at the centers of the four subrectangles shown on the next slide: Calculus

  24. Solution (cont’d) Calculus

  25. Solution (cont’d) • The area of each subrectangle is ΔA = ½, so Calculus

  26. Using More Subrectangles • If we keep dividing each subrectangle into four smaller ones, we get the Midpoint Rule approximations shown. • These valuesapproach the exactvalue of the doubleintegral, –12. Calculus

  27. Average Value • We define the average value of a function f of one variable defined on a rectangle R as where A(R) is the area of R. • If f(x, y) ≥ 0, the equation Calculus

  28. Average Value (cont’d) says that the box with base R and height fave has the same volume as the solid that lies under the graph of f. • If z = f(x, y) describes a mountainous region and we chop off the tops of the mountains at height fave, then we can use them to fill in the valleys so that the region becomes completely flat: Calculus

  29. Average Value (cont’d) Calculus

  30. Properties of Double Integrals • On the next slide we list three properties of double integrals. • We assume that all of the integrals exist. • The first two properties are referred to as the linearity of the integral: Calculus

  31. Properties (cont’d) • If f(x, y) ≥ g(x, y) for all (x, y) in R, then Calculus

  32. Review • Volumes and double integrals • Definition of double integral using Riemann sums • Midpoint Rule • Average value • Properties of double integrals Calculus

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