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 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

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 = = = 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 (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