1. Calculus I (MAT 145) Dr. Day Monday April 15, 2019
Chapter 5: Integrals and Anti-Derivartives
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”
Monday, April 15, 2019
MAT 145

3. If we know a rate function . . .
A particle moves along the x-axis. It’s velocity is given by
v(t) = 2t2-3t+1
If we know that the particle is at location s = 3 at time t = 0, that is, that s(0) = 3, determine the position function s(t). What is the particle’s location at time t = 10?
Monday, April 15, 2019
MAT 145

4. If we know a rate function . . .
Snow begins falling at midnight at a rate of 1 inch of snow per hour. It stops snowing at 6 am, 6 hours later. Write an accumulation function S(t), to describe the total amount of snow that had fallen by time t, where 0 ≤ t ≤ 6 hrs.
Monday, April 15, 2019
MAT 145

5. Accumulate, Accumulate, Accumulate!
How much snow fell?
Monday, April 15, 2019
MAT 145

6. Accumulate, Accumulate, Accumulate!
How much snow fell?
Monday, April 15, 2019
MAT 145

7. Accumulate, Accumulate, Accumulate!
How much snow fell?
Monday, April 15, 2019
MAT 145

8. Accumulate, Accumulate, Accumulate!
How much snow fell?
Monday, April 15, 2019
MAT 145

9. Accumulate, Accumulate, Accumulate!
How much snow fell?
Monday, April 15, 2019
MAT 145

10. Accumulate, Accumulate, Accumulate!
How much snow fell?
Monday, April 15, 2019
MAT 145

11. Accumulate, Accumulate, Accumulate!
How much snow fell?
Monday, April 15, 2019
MAT 145

12. Accumulate, Accumulate, Accumulate!
How much snow fell?
Monday, April 15, 2019
MAT 145

13. Areas and Distances (5.1)
Use What You Know to Get at What You’re Looking For
Choosing Endpoints
Notation
Accumulations From Rates
Monday, April 15, 2019
MAT 145

14. 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.
Monday, April 15, 2019
MAT 145

15. Monday, April 15, 2019
MAT 145

16. Approximating Area: Riemann Sums
Riemann Sum Applet
Monday, April 15, 2019
MAT 145

17. Monday, April 15, 2019
MAT 145

18. 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.
Monday, April 15, 2019
MAT 145

19. 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 ?
Monday, April 15, 2019
MAT 145

20. 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
Monday, April 15, 2019
MAT 145

21. Monday, April 15, 2019
MAT 145

22. Monday, April 15, 2019
MAT 145