1. Calculus II (MAT 146) Dr. Day Friday, September 01, 2017
Application #2: Average Value of a Function (Sec 6.5)
Application #3: Volumes of 3-dimensional solids (Secs 6.2, 6.3)
For Next Time . . .
Friday, September 1, 2017
MAT 146

2. 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, September 1, 2017
MAT 146

3. 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.
Friday, September 1, 2017
MAT 146

4. The average value of a function f, on a ≤ x ≤ b, with f continuous on that interval, is:
Friday, September 1, 2017
MAT 146

5. Determine the average value of y = x2 on [0,3].
Friday, September 1, 2017
MAT 146

6. Friday, September 1, 2017
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7. Determine the average value of the function here, for the specified interval. Determine a value c such that f(c) generates that average value.
Friday, September 1, 2017
MAT 146

8. Friday, September 1, 2017
MAT 146

9. Wednesday, February 1, 2017
MAT 146

10. Wednesday, February 1, 2017
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11. Wednesday, February 1, 2017
MAT 146

12. 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
Wednesday, February 1, 2017
MAT 146

13. 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)
Wednesday, February 1, 2017
MAT 146

14. Wednesday, February 1, 2017
MAT 146

15. 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.
Wednesday, February 1, 2017
MAT 146

16. Consider the first-quadrant region R with borders
y = sin(x) y = 0 and x = π/2
Sketch region R on the xy-plane.
Wednesday, February 1, 2017
MAT 146

17. 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.
Wednesday, February 1, 2017
MAT 146

18. 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:
Wednesday, February 1, 2017
MAT 146