1. Estimating area under a curve
4.6 Riemann Sums
Estimating area under a curve
(Left, Right, Midpoint and Trapezoid)

2. When we find the area under a curve by adding rectangles, the answer is called a Riemann sum.
The width of a rectangle is called a subinterval.
The entire interval is called the partition.
subinterval
partition
Subintervals do not all have to be the same size.

3. Example:
We could estimate the area under the curve by drawing rectangles touching at their left corners.
This is called the Left-hand Rectangular Approximation Method (LRAM).
Approximate area:

4. We could also use a Right-hand Rectangular Approximation Method (RRAM).
Approximate area:

5. Another approach would be to use rectangles that touch at the midpoint
Another approach would be to use rectangles that touch at the midpoint. This is the Midpoint Rectangular Approximation Method (MRAM).
In this example there are four subintervals.
As the number of subintervals increases, so does the accuracy.
Approximate area:

6. The exact area under this curve is .
With 8 subintervals:
Approximate area:
The exact area under this
curve is
width of subinterval

7. Inscribed rectangles are all below the curve:
Circumscribed rectangles are all above the curve:

8. Trapezoidal Approximations
Another way to approximate a definite integral is to use n trapezoids, as shown in Figure 4.42.
In the development of this method,
assume that f is continuous and
positive on the interval [a, b].
So, the definite integral
represents the area of the region
bounded by the graph of f and the
x-axis, from x = a to x = b.
Figure 4.42

9. Example:
Approximate the area under
the same curve using the Trapezoidal Approximation Method:
Averaging right and left rectangles gives us trapezoids:

10. Approximate the area under
the same curve using the Trapezoidal Approximation Method:

11. Example:
Use a trapezoidal sum with 4 subintervals to approximate:

12. Trapezoidal Rule: ( (b-a)/n = width of subinterval )
This gives us a better approximation than either left or right rectangles.
Not necessary to memorize this because rarely are the subintervals of equal width.

13. Example:
Estimate the area under the curve by finding the following:
A left Riemann Sum with n=4
A right Riemann Sum n=4
A Trapezoidal Sum n=4

14. Classwork Example:
Find the area under the curve using 8 trapezoids from x = 2 to x = 6.

15. Homework:
4.6 pg odd, 39.
Don’t do Simpson’s Rule.

16. Homework:
4.6 Kuta WS and MMM pg. 139

17. Calculus HWQ 12/8
Estimate the definite integral using a trapezoidal sum with n=4. No Calculator.

18. Pop Quiz 12/11/2012:
Use a midpoint Riemann sum to approximate the area under the curve with 4 subintervals.