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Chapter 6 The Definite Integral

Chapter 6 The Definite Integral. § 6.1. Antidifferentiation. Antidifferentiation. Finding Antiderivatives. EXAMPLE. Find all antiderivatives of the given function. Theorems of Antidifferentiation. The Indefinite Integral. Rules of Integration. Finding Antiderivatives. EXAMPLE.

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Chapter 6 The Definite Integral

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  1. Chapter 6The Definite Integral

  2. §6.1 Antidifferentiation

  3. Antidifferentiation

  4. Finding Antiderivatives EXAMPLE Find all antiderivatives of the given function.

  5. Theorems of Antidifferentiation

  6. The Indefinite Integral

  7. Rules of Integration

  8. Finding Antiderivatives EXAMPLE Determine the following.

  9. Finding Antiderivatives EXAMPLE Find the function f(x) for which and f(1) = 3.

  10. Antiderivatives in Application EXAMPLE A rock is dropped from the top of a 400-foot cliff. Its velocity at time t seconds is v(t) = -32t feet per second. (a) Find s(t), the height of the rock above the ground at time t. (b) How long will the rock take to reach the ground? (c) What will be its velocity when it hits the ground?

  11. §6.2 Areas and Riemann Sums

  12. Area Under a Graph

  13. Area Under a Graph In this section we will learn to estimate the area under the graph of f(x) from x = a to x = b by dividing up the interval into partitions (or subintervals), each one having width where n = the number of partitions that will be constructed. In the example below, n = 4. A Riemann Sum is the sum of the areas of the rectangles generated above.

  14. Riemann Sums to Approximate Areas EXAMPLE Use a Riemann sum to approximate the area under the graph f(x) on the given interval using midpoints of the subintervals SOLUTION The partition of -2 ≤ x ≤ 2 with n = 4 is shown below. The length of each subinterval is x1 x2 x3 x4 -2 2

  15. Riemann Sums to Approximate Areas CONTINUED Observe the first midpoint is units from the left endpoint, and the midpoints themselves are units apart. The first midpoint is x1 = -2 + = -2 + .5 = -1.5. Subsequent midpoints are found by successively adding midpoints: -1.5, -0.5, 0.5, 1.5 The corresponding estimate for the area under the graph of f(x) is So, we estimate the area to be 5 (square units).

  16. Approximating Area With Midpoints of Intervals CONTINUED

  17. Riemann Sums to Approximate Areas EXAMPLE Use a Riemann sum to approximate the area under the graph f(x) on the given interval using left endpoints of the subintervals SOLUTION The partition of 1 ≤ x ≤ 3 with n = 5 is shown below. The length of each subinterval is 1 1.4 1.8 2.2 2.6 3 x1 x2 x3 x4 x5

  18. Riemann Sums to Approximate Areas CONTINUED The corresponding Riemann sum is So, we estimate the area to be 15.12 (square units).

  19. Approximating Area Using Left Endpoints CONTINUED

  20. §6.3 Definite Integrals and the Fundamental Theorem

  21. The Definite Integral Δx = (b – a)/n, x1, x2, …., xn are selected points from a partition [a, b].

  22. Calculating Definite Integrals EXAMPLE Calculate the following integral. SOLUTION The figure shows the graph of the function f(x) = x + 0.5. Since f(x) is nonnegative for 0 ≤ x ≤ 1, the definite integral of f(x) equals the area of the shaded region in the figure below. 1 1 0.5

  23. Calculating Definite Integrals CONTINUED The region consists of a rectangle and a triangle. By geometry, Thus the area under the graph is 0.5 + 0.5 = 1, and hence

  24. The Definite Integral

  25. Calculating Definite Integrals EXAMPLE Calculate the following integral.

  26. The Fundamental Theorem of Calculus

  27. The Fundamental Theorem of Calculus EXAMPLE Use the Fundamental Theorem of Calculus to calculate the following integral. Use TI 83 to compute the definite integral: 1) put f(x) into y1 and graph. 2) 2ndtrace7 3) Enter lower limit and upper limit at the prompts.

  28. Area Under a Curve as an Antiderivative

  29. §6.4 Areas in the xy-Plane

  30. Properties of Definite Integrals

  31. Area Between Two Curves

  32. Finding the Area Between Two Curves EXAMPLE Find the area of the region between y = x2 – 3x and the x-axis (y = 0) from x = 0 to x = 4.

  33. Finding the Area Between Two Curves EXAMPLE Write down a definite integral or sum of definite integrals that gives the area of the shaded portion of the figure.

  34. §6.5 Applications of the Definite Integral

  35. Average Value of a Function Over an Interval

  36. Average Value of a Function Over an Interval EXAMPLE Determine the average value of f(x) = 1 – x over the interval -1 ≤ x ≤ 1.

  37. Average Value of a Function Over an Interval EXAMPLE (Average Temperature) During a certain 12-hour period the temperature at time t (measured in hours from the start of the period) was degrees. What was the average temperature during that period?

  38. Consumers’ Surplus

  39. Consumers’ Surplus EXAMPLE Find the consumers’ surplus for the following demand curve at the given sales level x.

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