1. Calculus II (MAT 146) Dr. Day Wednesday, January 24, 2018
Return Quiz #2
Questions: Area Between Curves (Sec 6.1)
Another Application of the Definite Integral: Average Value of a Function (6.5)
Quiz #3
For Next Time . . .
Wednesday, January 24, 2018
MAT 146

2. Integration Applications: Area Between Curves (6.1)
Wednesday, January 24, 2018
MAT 146

3. Wednesday, January 24, 2018
MAT 146

4. Area Between Curves
Calculate the area between the graphs of y = x2 + 2 and y = 1 – x for 0 ≤ x ≤ 1.
Wednesday, January 24, 2018
MAT 146

5. Calculate the area under the curve y = x2 + 2 for 0 ≤ x ≤ 1.
Area Under a Curve
Calculate the area under the curve y = x for 0 ≤ x ≤ 1.
Wednesday, January 24, 2018
MAT 146

6. Calculate the area under the curve y = 1 – x for 0 ≤ x ≤ 1.
Area Under a Curve
Calculate the area under the curve y = 1 – x for 0 ≤ x ≤ 1.
Wednesday, January 24, 2018
MAT 146

7. Wednesday, January 24, 2018
MAT 146

8. Wednesday, January 24, 2018
MAT 146

9. Area Between Curves: Strategies
Graph the functions in question and identify the number of bounded regions as well as which function is greater than the other for each region.
Determine the x-axis intervals (or y-axis intervals) for the bounded regions. The interval endpoints may be explicitly stated or can be determined using algebraic techniques, most typically by setting the two functions equal to each other.
Draw in a typical rectangle and determine its area. This provides essential information for the area integral you need to create.
For each bounded region, create a definite integral to represent the sum of the areas of an infinite number of typical rectangles. Evaluate this integral to determine the area of each bounded region.
Note that your TI-89 or other CAS can be a useful tool for several components of your solution process.
Wednesday, January 24, 2018
MAT 146

10. Area Between Curves
(A) Calculate the first-quadrant area between the graphs of y = √x and y = x2. Show a picture of the enclosed region. (B) Set up one or more definite integrals to represent the finite area of the region enclosed by the graphs of y = 4x + 16 and y = 2x for−2 ≤ x ≤ 5. Do not calculate! (C) Determine the exact area of the region enclosed by the graphs of x = −y and x = (y – 2)2. Sketch a graph of the region.
Wednesday, January 24, 2018
MAT 146

11. Area Between Curves
1. Calculate the area between the graphs of y = 2x3 – 1 and y = x – 1 for 1 ≤ x ≤ 2.
2. Calculate the area between the graphs of y = (x–1)2 and y = 3 – x.
3. Calculate the area between the graphs of y = x3 –3x + 2 and y = x + 2.
4. Calculate the area between the graphs of x = y2 –1 and x = 3.
Wednesday, January 24, 2018
MAT 146

12. Area Between Curves
Calculate the area between the graphs of y = 2x3 – 1 and y = x – 1 for 1 ≤ x ≤ 2.
Wednesday, January 24, 2018
MAT 146

13. Wednesday, January 24, 2018
MAT 146

14. Calculate the area between the graphs of y = (x–1)2 and y = 3 – x.
Area Between Curves
Calculate the area between the graphs of y = (x–1)2 and y = 3 – x.
Wednesday, January 24, 2018
MAT 146

15. Wednesday, January 24, 2018
MAT 146

16. Calculate the area between the graphs of y = x3 –3x + 2 and y = x + 2.
Area Between Curves
Calculate the area between the graphs of y = x3 –3x + 2 and y = x + 2.
Wednesday, January 24, 2018
MAT 146

17. Wednesday, January 24, 2018
MAT 146

18. Calculate the area between the graphs of x = y2 –1 and x = 3.
Area Between Curves
Calculate the area between the graphs of x = y2 –1 and x = 3.
Wednesday, January 24, 2018
MAT 146

19. Wednesday, January 24, 2018
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20. Wednesday, January 24, 2018
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21. Average Values
Question: What was the average temperature between midnight and noon yesterday?
Wednesday, January 24, 2018
MAT 146

22. If all we know is that it was 25º F. at midnight and that it was 41º F
If all we know is that it was 25º F. at midnight and that it was 41º F. at noon, the average temperature is:
Wednesday, January 24, 2018
MAT 146

23. If we also know that it was 36º F
If we also know that it was 36º F. at 6 am, we can re-compute an average temperature:
Wednesday, January 24, 2018
MAT 146

24. What if we know hourly readings?
12 midnight
25° F
4 am
33° F
9 am
38° F
1 am
27° F
5 am
34° F
10 am
39° F
2 am
29° F
6 am
36° F
11 am
40° F
3am
31° F
7 am
37° F
12 noon
41° F
8 am
Wednesday, January 24, 2018
MAT 146

25. Wednesday, January 24, 2018
MAT 146

26. What if we know temperatures every minute? Every second?
Wednesday, January 24, 2018
MAT 146

27. The average value of a function f, on a ≤ x ≤ b, with f continuous on that interval, is:
Wednesday, January 24, 2018
MAT 146

28. Determine the average value of y = x2 on [0,3].
Wednesday, January 24, 2018
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29. Wednesday, January 24, 2018
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30. Determine the average value of the function here, for the specified interval. Determine a value c such that f(c) generates that average value.
Wednesday, January 24, 2018
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31. Wednesday, January 24, 2018
MAT 146