1. Calculus II (MAT 146) Dr. Day Wednesday, August 23, 2017
Integration Technique #1: U-Substitution (Sec 5.5)
Application #1: Area Between Curves (Sec 6.1)
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Wednesday, August 23, 2017
MAT 146

2. 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.
Wednesday, August 23, 2017
MAT 146

3. Integration Techniques: Substitution
Wednesday, August 23, 2017
MAT 146

4. Integration Applications: Area Between Curves (6.1)
Wednesday, August 23, 2017
MAT 146

5. Wednesday, August 23, 2017
MAT 146

6. Area Between Curves
Calculate the area between the graphs of y = x2 + 2 and y = 1 – x for 0 ≤ x ≤ 1.
Wednesday, August 23, 2017
MAT 146

7. 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, August 23, 2017
MAT 146

8. 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, August 23, 2017
MAT 146

9. Area Between Curves
Calculate the area between the graphs of y = x2 + 2 and y = 1 – x for 0 ≤ x ≤ 1.
Wednesday, August 23, 2017
MAT 146

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MAT 146

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MAT 146

12. 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 23, 2017
MAT 146

13. Area Between Curves
Calculate the area between the graphs of y = 2x3 – 1 and y = x – 1 for 1 ≤ x ≤ 2.
Wednesday, August 23, 2017
MAT 146

14. Wednesday, August 23, 2017
MAT 146

15. 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 23, 2017
MAT 146

16. Wednesday, August 23, 2017
MAT 146

17. 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 23, 2017
MAT 146

18. Wednesday, August 23, 2017
MAT 146

19. 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 23, 2017
MAT 146

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MAT 146

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MAT 146

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MAT 146

23. 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 23, 2017
MAT 146

24. 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 23, 2017
MAT 146

25. Making Connections CHANGE ACCUMULATE CALCULUS! PRE-CALCULUS! LIMITS
FUNCTIONS
ACCUMULATE
PRE-CALCULUS!
Wednesday, August 23, 2017
MAT 146

26. Big Ideas From Calc I
Wednesday, August 23, 2017
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MAT 146