§ 6.2 Areas and Riemann Sums

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

Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #18 Area Under a Graph DefinitionExample 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 #19 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 #20 Riemann Sums to Approximate AreasEXAMPLE SOLUTION Use a Riemann sum to approximate the area under the graph f (x) on the given interval using midpoints of the subintervals The partition of -2 x 2 with n = 4 is shown below. The length of each subinterval is -22 x1x1 x2x2 x3x3 x4x4

Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #21 Riemann Sums to Approximate Areas Observe the first midpoint is units from the left endpoint, and the midpoints themselves are units apart. The first midpoint is x 1 = -2 + = = Subsequent midpoints are found by successively adding CONTINUED 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 #22 Approximating Area With Midpoints of IntervalsCONTINUED

Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #23 Riemann Sums to Approximate AreasEXAMPLE SOLUTION Use a Riemann sum to approximate the area under the graph f (x) on the given interval using left endpoints of the subintervals The partition of 1 x 3 with n = 5 is shown below. The length of each subinterval is 3 x1x1 x2x2 x3x3 x4x4 x5x

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

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

Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #26 Applications of Approximating AreasEXAMPLE SOLUTION 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.) Since measurements of the cars velocity were taken every ten seconds, we will use. Now, upon seeing the graph of the cars velocity, we can construct a Riemann sum to estimate how far the car traveled.

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

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