1. Calculus II (MAT 146) Dr. Day Friday, January 19, 2018
Integration Technique #1: U-Substitution (Sec 5.5)
Application #1: Area Between Curves (Sec 6.1)
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Friday, January 19, 2018
MAT 146

2. U-Substitution (5.5)
More Practice!
Friday, January 19, 2018
MAT 146

3. Evaluate each integral using u-substitution
Evaluate each integral using u-substitution. Clearly indicate your connection between u and x. Include notation for the differentials du and dx throughout.
Friday, January 19, 2018
MAT 146

4. Integration Techniques: Substitution
Friday, January 19, 2018
MAT 146

5. Integration Applications: Area Between Curves (6.1)
Friday, January 19, 2018
MAT 146

6. Friday, January 19, 2018
MAT 146

7. Area Between Curves
Calculate the area between the graphs of y = x2 + 2 and y = 1 – x for 0 ≤ x ≤ 1.
Friday, January 19, 2018
MAT 146

8. 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.
Friday, January 19, 2018
MAT 146

9. 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.
Friday, January 19, 2018
MAT 146

10. Area Between Curves
Calculate the area between the graphs of y = x2 + 2 and y = 1 – x for 0 ≤ x ≤ 1.
Friday, January 19, 2018
MAT 146

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MAT 146

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MAT 146

13. 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.
Friday, January 19, 2018
MAT 146

14. Area Between Curves
Calculate the area between the graphs of y = 2x3 – 1 and y = x – 1 for 1 ≤ x ≤ 2.
Friday, January 19, 2018
MAT 146

15. Friday, January 19, 2018
MAT 146

16. 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.
Friday, January 19, 2018
MAT 146

17. Friday, January 19, 2018
MAT 146

18. 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.
Friday, January 19, 2018
MAT 146

19. Friday, January 19, 2018
MAT 146

20. 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.
Friday, January 19, 2018
MAT 146

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MAT 146

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MAT 146

24. 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 19, 2018
MAT 146

25. 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, January 19, 2018
MAT 146

26. Making Connections CHANGE ACCUMULATE CALCULUS! PRE-CALCULUS! LIMITS
FUNCTIONS
ACCUMULATE
PRE-CALCULUS!
Friday, January 19, 2018
MAT 146

27. Big Ideas From Calc I
Friday, January 19, 2018
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28. Friday, January 19, 2018
MAT 146