1. Definite Integrals & Riemann Sums

2. Definition of a Definite Integral
A definite integral of a function y = f(x) on an interval [a, b] is the signed () area between the curve and the x-axis.

3. Example1
Find an approximate value for the area of the shaded region A1
A  2

4. Example 2
Find an approximate value for the area of the shaded region A2
A  4.2

5. Example 3
Find an approximate value for the area of the shaded region A1 + A2
A  6.2

6. What is a Riemann Sum?
A Riemann Sum is a series of rectangles that are used to approximate the definite integral in a specific way.
Visualization of Riemann Sums

7. How is a Riemann Sum related to a definite integral?
Approximate the area under the curve of f(x) = 2x – 2x2 on the interval [0,1] using 10 rectangles.

8. A Better Approximation:
Let the number of rectangles approach ∞
Then for a function f(x) on an interval [a, b] where f(x) ≥ 0 the area under the curve is
Which is the same as

9. Left Riemann Sum – the long way
Example: Find the Left Riemann Sum for the function
y = x on the interval [-1, 2] using 6 partitions of equal width.
Determine the width of each subinterval.
Sketch out the function.

10. Left Riemann Sum- the short way when uniform width
Example: Find the Left Riemann Sum for the function
y = x on the interval [-1, 2] using 6 partitions of equal width. TI - 89
Determine the width of each subinterval.
Left Sum – Start at the left-most endpoint and find all of the corresponding y-values for each subinterval except the right-most endpoint.
Add the y-values and multiply by the partition width.

11. Left Riemann Sum The following Riemann Sum approximates the integral
Δx = 3/6 or .5
Need f(-1), f(-.5), f(0), f(.5), f(1) and f(1.5)
Then,

12. Right Riemann Sum Example: Find the Right Riemann Sum for the function
y = x on the interval [-1, 2] using 6 partitions of equal width.
Determine the width of each subinterval.
Right Sum – Start at the right-most endpoint and find all of the corresponding y-values for each subinterval except the left-most endpoint.
Add the y-values and multiply by the partition width.

13. Right Riemann Sum The following Riemann Sum approximates the integral
Δx = 3/6 or .5
Need f(-.5), f(0), f(.5), f(1), f(1.5) and f(2)
Then,

14. Using Sums to Approximate Area
When using Riemann Sums to approximate the area between a curve and the x-axis, take the absolute value of any y-value that is negative.
Example: Use a Left Riemann Sum to approximate the area between the curve
y = 1 – x2 and the x-axis from [0, 2] using 4 equal partitions. TI - 89

15. Solution - Area 1. Δx = 2/4 = .5 Need f(0), f(.5), f(1), f(1.5)
Take absolute value of any negative y-values (f(1.5))
Then,

16. AP - Only What if the partitions are of unequal width?
Example: 2009 Q5 part c
Use a left Riemann sum with subintervals indicated by the data in the table to approximate x 2 3 5 8 13
f(x) 1 4 -2 6

17. The Midpoint Rule The midpoint rule is a Riemann Sum that
uses the midpoint of each subinterval
as the height of each rectangle. 2

18. Midpoint Riemann Sum
Example: Use the Midpoint Rule to estimate the integral
using 6 partitions of equal width.
Determine the width of each subinterval.
Find the midpoint of each subinterval and their corresponding y-values.
Add the y-values and multiply by the partition width.

19. Midpoint Riemann Sum Δx = 3/6 or .5
Midpoints occur at f(-.75), f(-.25), f(.25), f(.75), f(1.25), f(1.75)
Then,

20. Upper and Lower Riemann Sums
Example: Find both the Lower and Upper Riemann Sums for the function
y = x2 + 1 from [-2, 2] using 4 equal partitions.

21. Upper and Lower Sums
Example: Find both the Lower and Upper Riemann Sums for the function
y = 1 – x2 from [-1, 2] using 6 equal partitions.

22. Upper Riemann Sum Δx = 3/6 or .5
Use f(-.5), f(0), f(0), f(.5), f(1), f(1.5)
Then,

23. Lower Riemann Sum Δx = 3/6 or .5
Use f(-1), f(-.5), f(.5), f(1), f(1.5), f(2)
Then,