§ 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 Definition Example Area Under the Graph of f (x) from a to b: An example of this is shown to the right Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #18

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

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

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

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

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

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

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

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

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

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