1. Calculus II (MAT 146) Dr. Day Friday, January 26, 2018
Return Quiz #3 and Extended Homework A
Integral Application #2: Average Value of a Function (6.5)
Integral Application #3: Volumes of Solids (6.2 and 6.3)
For Next Time . . .
Friday, January 26, 2018
MAT 146

2. Friday, January 26, 2018
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3. Friday, January 26, 2018
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4. 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.
Friday, January 26, 2018
MAT 146

5. Average Values
Question: What was the average temperature between midnight and noon yesterday?
Friday, January 26, 2018
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6. 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:
Friday, January 26, 2018
MAT 146

7. 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:
Friday, January 26, 2018
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8. 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
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10. What if we know temperatures every minute? Every second?
Friday, January 26, 2018
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11. The average value of a function f, on a ≤ x ≤ b, with f continuous on that interval, is:
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12. Determine the average value of y = x2 on [0,3].
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14. Determine the average value of the function here, for the specified interval. Determine a value c such that f(c) generates that average value.
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19. Region R in the first quadrant of the xy-plane is bordered by the x-axis, the line x = 4, and the curve y = √x.
Determine the volume of the solid of revolution generated when R is rotated about the line y = 2.
Determine the volume of the solid of revolution generated when R is rotated about the line x = −1.
(A) (8pi)/3 (B) (544pi)/15
Friday, January 26, 2018
MAT 146

20. Volumes of Solids of Revolution (6.2 & 6.3)
Dynamic Illustration #1 (discs)
Dynamic Illustration #2 (washer)
Dynamic Illustration #3 (shell)
Dynamic Illustration #4 (cross section I) (cross section II)
Friday, January 26, 2018
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21. Friday, January 26, 2018
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22. Consider the first-quadrant region R with borders
y = sin(x) y = 0 and x = π/2
Sketch region R on the xy-plane.
Calculate the exact area of R. Show evidence to support your solution.
Set up, but do not calculate, a definite integral to represent the volume of the solid created when R is revolved around the y-axis.
Friday, January 26, 2018
MAT 146

23. Consider the first-quadrant region R with borders
y = sin(x) y = 0 and x = π/2
Sketch region R on the xy-plane.
Friday, January 26, 2018
MAT 146

24. Consider the first-quadrant region R with borders
y = sin(x) y = 0 and x = π/2
Calculate the exact area of R. Show evidence to support your solution.
Friday, January 26, 2018
MAT 146

25. Consider the first-quadrant region R with borders
y = sin(x) y = 0 and x = π/2
Set up, but do not calculate, a definite integral to represent the volume of the solid created when R is revolved around the y-axis.
Shells:
Washers:
Friday, January 26, 2018
MAT 146