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§ 6.2

§ 6.2. Areas and Riemann Sums. Section Outline. Area Under a Graph Riemann Sums to Approximate Areas (Midpoints) Riemann Sums to Approximate Areas (Left Endpoints) Applications of Approximating Areas. Area Under a Graph.

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§ 6.2

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  1. §6.2 Areas and Riemann Sums

  2. Section Outline • Area Under a Graph • Riemann Sums to Approximate Areas (Midpoints) • Riemann Sums to Approximate Areas (Left Endpoints) • Applications of Approximating Areas

  3. Area Under a Graph Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #18

  4. 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. Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #19

  5. 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 Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #20

  6. 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). Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #21

  7. Approximating Area With Midpoints of Intervals CONTINUED Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #22

  8. 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 Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #23

  9. Riemann Sums to Approximate Areas CONTINUED The corresponding Riemann sum is So, we estimate the area to be 15.12 (square units). Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #24

  10. Approximating Area Using Left Endpoints CONTINUED Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #25

  11. Applications of Approximating Areas EXAMPLE The velocity of a car (in feet per second) is recorded from the speedometer every 10 seconds, beginning 5 seconds after the car starts to move. See Table 2. Use a Riemann sum to estimate the distance the car travels during the first 60 seconds. (Note: Each velocity is given at the middle of a 10-second interval. The first interval extends from 0 to 10, and so on.) SOLUTION Since measurements of the car’s velocity were taken every ten seconds, we will use . Now, upon seeing the graph of the car’s velocity, we can construct a Riemann sum to estimate how far the car traveled. Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #26

  12. Applications of Approximating Areas CONTINUED Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #27

  13. Applications of Approximating Areas CONTINUED Therefore, we estimate that the distance the car traveled is 2800 feet. Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #28

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