1. Calculus I (MAT 145) Dr. Day Wednesday April 17, 2019
Chapter 5: Integrals and Anti-Derivatives
Recovering a Function Knowing its Derivative
Riemann Sums
Riemann Sums with an Infinite Number of Subdivisions
Definite Integrals and Indefinite Integrals: Connecting Derivatives and Anti- Derivatives
The Fundamental Theorem of Calculus
Part 1
Part 2
Monday, April 15, 2019
MAT 145

2. Antiderivatives, Integrals, and Initial-Value Problems
Knowing f ’, can we determine f ? General and specific solutions:
The antiderivative.
The integral symbol: Representing antiderivatives
Initial-Value Problems: Transforming a general antiderivative into a specific function that satisfies the given information.
Read this:
“The antiderivative
of 2x
with respect to the variable x”
Wednesday, April 17, 2019
MAT 145

3. Accumulate, Accumulate, Accumulate!
How much snow fell?
Wednesday, April 17, 2019
MAT 145

4. Accumulate, Accumulate, Accumulate!
How much snow fell?
Wednesday, April 17, 2019
MAT 145

5. Accumulate, Accumulate, Accumulate!
How much snow fell?
Wednesday, April 17, 2019
MAT 145

6. Accumulate, Accumulate, Accumulate!
How much snow fell?
Wednesday, April 17, 2019
MAT 145

7. Accumulate, Accumulate, Accumulate!
How much snow fell?
Wednesday, April 17, 2019
MAT 145

8. Accumulate, Accumulate, Accumulate!
How much snow fell?
Wednesday, April 17, 2019
MAT 145

9. Areas and Distances (5.1)
Use What You Know to Get at What You’re Looking For
Choosing Endpoints
Notation
Accumulations From Rates
Wednesday, April 17, 2019
MAT 145

10. Approximating Area: Riemann Sums
To generate a way to calculate the area under the curve of a rate function, in order to determine an accumulation, we begin with AREA APPROXIMATIONS. We create something called a Riemann Sum and use better and better area approximations that will lead to exact area.
Wednesday, April 17, 2019
MAT 145

11. Approximating Area: Riemann Sums
Riemann Sum Applet
Wednesday, April 17, 2019
MAT 145

12. Wednesday, April 17, 2019
MAT 145

13. Riemann Sums
Calculate the exact value of a Riemann Sum to approximate the area under the curve y = x2-4 for 1 ≤ x ≤ 4, using n = 3 rectangles and using midpoints.
Show a graph of the function that includes a sketch of your rectangles.
Show all calculations using exact values.
Clearly indicate the value of your Riemann Sum.
Wednesday, April 17, 2019
MAT 145

14. Riemann Sums
Suppose y = f(x) = x2-4 is an object’s velocity function. On 1 ≤ x ≤ 4:
What is the NET CHANGE IN POSITION?
What is the TOTAL DISTANCE TRAVELED?
If 𝐹 𝑡 = 1 𝑡 𝑓 𝑥 𝑑𝑥 , What is 𝐹′ 𝑡 ?
Wednesday, April 17, 2019
MAT 145

15. upper limit of integration
differential
This is called a definite integral. It includes lower bounds and upper bounds that represent boundary values of an x-axis interval.
integrand
lower limit of integration
Wednesday, April 17, 2019
MAT 145

16. Wednesday, April 17, 2019
MAT 145

17. Wednesday, April 17, 2019
MAT 145

18. Try These!
Wednesday, April 17, 2019
MAT 145

19. The Fundamental Theorem of Calculus (Part II)
Let f be continuous on [a, b]. Then,
where F is any antiderivative of f; that is, F ′(x) = f(x).
Wednesday, April 17, 2019
MAT 145

20. The Fundamental Theorem of Calculus (Part II)
Wednesday, April 17, 2019
MAT 145

21. The Fundamental Theorem of Calculus (Part II)
Wednesday, April 17, 2019
MAT 145

22. The Fundamental Theorem of Calculus (Part II)
Wednesday, April 17, 2019
MAT 145

23. The Fundamental Theorem of Calculus (Part I)
For f continuous on [a, b], let the function g be
Then g(x) is an antiderivative of f:
Wednesday, April 17, 2019
MAT 145

24. Rates and Accumulations
Water flows out of a tank at liters per second, t ≥ 0.
(1) How much water left the tank during the first 2 minutes of drainage?
(2) If the tank contained 6000 liters of water when the outflow began, how much water was in the tank after 5 minutes of drainage?
(3) If the tank contained 6000 liters of water when the outflow began, how much time is required to drain the tank? (nearest second)
Wednesday, April 17, 2019
MAT 145

25. Rates and Accumulations
Water flows out of a tank at liters per second, t ≥ 0.
How much water left the tank during the first 2 minutes of drainage?
886 2/3 liters of water left the tank during the first two minutes.
Wednesday, April 17, 2019
MAT 145

26. Rates and Accumulations
Water flows out of a tank at liters per second, t ≥ 0.
(2) If the tank contained 6000 liters of water when the outflow began, how much water was in the tank after 5 minutes of drainage?
There are liters lefts in the tank after 5 minutes.
Wednesday, April 17, 2019
MAT 145

27. Rates and Accumulations
Water flows out of a tank at liters per second, t ≥ 0.
(3) If the tank contained 6000 liters of water when the outflow began, how much time is required to drain the tank? (nearest second)
It will take about 432 seconds to drain the tank.
Wednesday, April 17, 2019
MAT 145

28. Rates and Accumulations
The velocity function (in meters per second) is given for a particle moving along a line.
Calculate the displacement for the given time interval. (Note: Displacement is difference between starting position and ending position.)
The displacement between t=0 and t=4 is 0 units.
Wednesday, April 17, 2019
MAT 145