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) Admin Questions (Course Requirements/Expectations) ? For Next Time . . . Wednesday, August 23, 2017 MAT 146

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

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

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

Wednesday, August 23, 2017 MAT 146

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

Calculate the area under the curve y = x2 + 2 for 0 ≤ x ≤ 1. Area Under a Curve Calculate the area under the curve y = x2 + 2 for 0 ≤ x ≤ 1. Wednesday, August 23, 2017 MAT 146

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

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

Wednesday, August 23, 2017 MAT 146

Wednesday, August 23, 2017 MAT 146

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

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

Wednesday, August 23, 2017 MAT 146

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

Wednesday, August 23, 2017 MAT 146

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

Wednesday, August 23, 2017 MAT 146

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

Wednesday, August 23, 2017 MAT 146

Wednesday, August 23, 2017 MAT 146

Wednesday, August 23, 2017 MAT 146

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 = 2x2 + 10 for−2 ≤ x ≤ 5. Do not calculate! (C) Determine the exact area of the region enclosed by the graphs of x = −y2 + 10 and x = (y – 2)2. Sketch a graph of the region. Wednesday, August 23, 2017 MAT 146

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

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

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

Wednesday, August 23, 2017 MAT 146