1. Chapter 5 – Integrals 5.1 Areas and Distances Dr. Erickson

2. Consider an object moving at a constant rate of 3 ft/sec.
Since rate . time = distance:
If we draw a graph of the velocity, the distance that the object travels is equal to the area under the line.
time
velocity
After 4 seconds, the object has gone 12 feet.
5.1 Areas and Distances
Dr. Erickson

3. Velocity is not Constant
If the velocity is not constant,
we might guess that the
distance traveled is still equal
to the area under the curve.
(The units work out.)
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 (L).
Approximate area:
5.1 Areas and Distances
Dr. Erickson

4. Right-hand Rectangular Approximation Method (R).
Approximate area:
5.1 Areas and Distances
Dr. Erickson

5. Definition #2 – Using Right End Points
The area A of the region S that lies under the graph of the continuous function f is the limit of the sum of the areas of approximating rectangles:
5.1 Areas and Distances
Dr. Erickson

6. Definition Using Left Endpoints
We get the same values if we use left endpoints.
5.1 Areas and Distances
Dr. Erickson

7. Midpoint Rectangular Approximation Method (M)
Another approach would be to use rectangles that touch at the midpoint. This is the Midpoint Rectangular Approximation Method.
In this example there are four subintervals.
As the number of subintervals increases, so does the accuracy.
Approximate area:
5.1 Areas and Distances
Dr. Erickson

8. width of subinterval With 8 subintervals: Approximate area:
The exact answer for this
problem is
width of subinterval
5.1 Areas and Distances
Dr. Erickson

9. Definition Using Midpoints
We can consider the midpoint to be sample points ( ) so we have the formula:
5.1 Areas and Distances
Dr. Erickson

10. Sigma Notation
We often use sigma notation to write the sums with many terms more compactly. For instance,
5.1 Areas and Distances
Dr. Erickson

11. Example 1
Estimate the area under the graph from x=0 to x=4 using four approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or overestimate?
Repeat part (a) using left endpoints.
5.1 Areas and Distances
Dr. Erickson

12. Example 2 Graph the function.
Estimate the area under the graph from x=-2 to x=2 using four approximating rectangles and taking sample points to be (i) right endpoints and (ii) midpoints. In each case, sketch the graph and the rectangles.
Improve your estimates in part (b) using 8 rectangles.
5.1 Areas and Distances
Dr. Erickson

13. Example 3
Oil leaked from a tank at a rate of r(t) liters per hour. The rate decreased as time passed and values of the rate at two-hour time intervals are shown in the table. Find the lower and upper estimates for the total amount of oil that leaked out.
t(h) 2 4 6 8 10
r(t) (L/h)
8.7
7.6
6.8
6.2
5.7
5.3
5.1 Areas and Distances
Dr. Erickson

14. Example 4
Use the definition (#2 according to your book) to find an expression for the area under the graph of f as a limit. Do not evaluate the limit. a) b)
5.1 Areas and Distances
Dr. Erickson

15. Example 5
Determine a region whose area is equal to the given limit. Do not evaluate the limit.
5.1 Areas and Distances
Dr. Erickson