1. Sections 4.2-4.4: Area, Definite Integrals, and The Fundamental Theorem of Calculus

2. Applying the well known formula:
Distance Traveled
A train moves along a track at a steady rate of 75 miles per hour from 3:00 A.M. to 8:00 A.M. What is the total distance traveled by the train?
Applying the well known formula:
Velocity (mph)
Notice the distance traveled by the train (375 miles) is exactly the area of the rectangle whose base is the time interval [3,8] and whose height is the constant velocity function v=75.
Time (h)

3. Distance Traveled
A particle moves along the x-axis with velocity v(t)=-t2+2t+5 for time t≥0 seconds. How far is the particle after 3 seconds?
Velocity
The distance traveled is still the area under the curve. Unfortunately the shape is a irregular region. We need to find a method to find this area.
Time

4. The Area Problem
We now investigate how to solve the area problem: Find the area of the region S that lies under the curve y=f(x) from a to b.
f(x)
This means S is bounded by the graph of a continuous function, two vertical lines, and the x-axis. S a b

5. Finding Area
It is easy to calculate the area of certain shapes because familiar formulas exist:
A=½bh
A=lw
The area of irregular polygons can be found by dividing them into convenient shapes and their areas: A2 A3 A1 A4

6. Approximating the Area Under a Curve
We first approximate the area under a function by rectangles. a b

7. Approximating the Area Under a Curve
Then we take the limit of the areas of these rectangles as we increase the number of rectangles. a b

8. Approximating the Area Under a Curve
Then we take the limit of the areas of these rectangles as we increase the number of rectangles. a b

9. Estimating Area Using Rectangles and Right Endpoints
Use rectangles to estimate the area under the parabola y=x2 from 0 to 1 using 4 rectangles and right endpoints.
Make rectangles whose base is the same as the strip and whose height is the same as the right edge of the strip.
Width = ¼ and height = value of the function at ¼
Find the Sum of the Areas:
Divide the area under the curve into 4 equal strips

10. Estimating Area Using Rectangles and Right Endpoints
Use rectangles to estimate the area under the parabola y=x2 from 0 to 1 using 8 rectangles and right endpoints.
Make rectangles whose base is the same as the strip and whose height is the same as the right edge of the strip.
Width = 1/8 and height = value of the function at 1/8
Find the Sum of the Areas:
Divide the area under the curve into 8 equal strips

11. Estimating Area Using Rectangles and Left Endpoints
Use rectangles to estimate the area under the parabola y=x2 from 0 to 1 using 4 rectangles and left endpoints.
Make rectangles whose base is the same as the strip and whose height is the same as the left edge of the strip.
Width = ¼ and height = value of the function at 0
Find the Sum of the Areas:
Divide the area under the curve into 4 equal strips

12. Estimating Area Using Rectangles and Left Endpoints
Use rectangles to estimate the area under the parabola y=x2 from 0 to 1 using 8 rectangles and left endpoints.
Make rectangles whose base is the same as the strip and whose height is the same as the left edge of the strip.
Width = 1/8 and height = value of the function at 0
Find the Sum of the Areas:
Divide the area under the curve into 8 equal strips

13. Distance Traveled
A particle moves along the x-axis with velocity v(t)=-t2+2t+5 for time t≥0 seconds. Use three midpoint rectangles to estimate how far the particle traveled after 3 seconds?
Width = 1 and height = value of the function at 0.5
Make rectangles whose base is the same as the strip and whose height is the same as the middle of the strip.
Velocity
Find the Sum of the Areas:
Divide the area under the curve into 3 equal strips
Time

14. Negative Area
If a function is less than zero for an interval, the region between the graph and the x-axis represents negative area.
Positive Area
Negative Area

15. Definite Integral: Area Under a Curve
If y=f(x) is integrable over a closed interval [a,b], then the area under the curve y=f(x) from a to b is the integral of f from a to b.
Upper limit of integration
Lower limit of integration

16. The Existence of Definite Integrals
All continuous functions are integrable. That is, if a function f is continuous on an interval [a,b], then its definite integral over [a,b] exists . Ex:

17. Rules for Definite Integrals
Let f and g be functions and x a variable; a, b, c, and k be constant.
Constant Multiple
Sum Rule
Difference Rule
Additivity

18. The First Fundamental Theorem of Calculus
If f is continuous on the interval [a,b] and F is any function that satisfies F '(x) = f(x) throughout this interval then

19. Example 1
Evaluate
First Find the indefinte integral F(x):
Now apply the FTC to find the definite integral:
Notice that it is not necessary to include the “C” with definite integrals

20. Examples: New Notation
1. Evaluate
F(x)
Bounds
2. Evaluate
3. Evaluate