1. Calculus II (MAT 146) Dr. Day Wednesday, August 30, 2017
Return Quiz #3
Application #1: Area Between Curves (Sec 6.1)
Application #2: Average Value of a Function (Sec 6.5)
For Next Time . . .
Wednesday, August 30, 2017
MAT 146

2. Integration Applications: Area Between Curves (6.1)
Wednesday, August 30, 2017
MAT 146

3. Wednesday, August 30, 2017
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, August 30, 2017
MAT 146

5. Wednesday, August 30, 2017
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6. Wednesday, August 30, 2017
MAT 146

7. 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, August 30, 2017
MAT 146

8. Area Between Curves
Calculate the area between the graphs of y = 2x3 – 1 and y = x – 1 for 1 ≤ x ≤ 2.
Wednesday, August 30, 2017
MAT 146

9. Wednesday, August 30, 2017
MAT 146

10. 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, August 30, 2017
MAT 146

11. Wednesday, August 30, 2017
MAT 146

12. 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, August 30, 2017
MAT 146

13. Wednesday, August 30, 2017
MAT 146

14. 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, August 30, 2017
MAT 146

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MAT 146

16. Wednesday, August 30, 2017
MAT 146

17. 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, August 30, 2017
MAT 146

18. 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, August 30, 2017
MAT 146

19. Average Values
Question: What was the average temperature between midnight and noon yesterday?
Wednesday, August 30, 2017
MAT 146

20. If all we know is that it was 70º F. at midnight and that it was 86º F
If all we know is that it was 70º F. at midnight and that it was 86º F. at noon, the average temperature is:
Wednesday, August 30, 2017
MAT 146

21. If we also know that it was 81º F
If we also know that it was 81º F. at 6 am, we can re-compute an average temperature:
Wednesday, August 30, 2017
MAT 146

22. What if we know hourly readings?
12 midnight
70° F
4 am
78° F
9 am
83° F
1 am
72° F
5 am
79° F
10 am
84° F
2 am
74° F
6 am
81° F
11 am
85° F
3am
75° F
7 am
82° F
12 noon
86° F
8 am
Wednesday, August 30, 2017
MAT 146

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24. What if we know temperatures every minute? Every second?
Wednesday, August 30, 2017
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25. The average value of a function f, on a ≤ x ≤ b, with f continuous on that interval, is:
Wednesday, August 30, 2017
MAT 146

26. Determine the average value of y = x2 on [0,3].
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28. 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, August 30, 2017
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