Calculus II (MAT 146) Dr. Day Wednesday, August 30, 2017 Return Quiz #3 Application #1: Area Between Curves (Sec 6.1) Application #2: Average Value of a Function (Sec 6.5) For Next Time . . . Wednesday, August 30, 2017 MAT 146

Integration Applications: Area Between Curves (6.1) Wednesday, August 30, 2017 MAT 146

Wednesday, August 30, 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 30, 2017 MAT 146

Wednesday, August 30, 2017 MAT 146

Wednesday, August 30, 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 30, 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 30, 2017 MAT 146

Wednesday, August 30, 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 30, 2017 MAT 146

Wednesday, August 30, 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 30, 2017 MAT 146

Wednesday, August 30, 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 30, 2017 MAT 146

Wednesday, August 30, 2017 MAT 146

Wednesday, August 30, 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 30, 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 30, 2017 MAT 146

Average Values Question: What was the average temperature between midnight and noon yesterday? Wednesday, August 30, 2017 MAT 146

If all we know is that it was 70º F. at midnight and that it was 86º F If all we know is that it was 70º F. at midnight and that it was 86º F. at noon, the average temperature is: Wednesday, August 30, 2017 MAT 146

If we also know that it was 81º F If we also know that it was 81º F. at 6 am, we can re-compute an average temperature: Wednesday, August 30, 2017 MAT 146

What if we know hourly readings? 12 midnight 70° F 4 am 78° F 9 am 83° F 1 am 72° F 5 am 79° F 10 am 84° F 2 am 74° F 6 am 81° F 11 am 85° F 3am 75° F 7 am 82° F 12 noon 86° F 8 am Wednesday, August 30, 2017 MAT 146

Wednesday, August 30, 2017 MAT 146

What if we know temperatures every minute? Every second? Wednesday, August 30, 2017 MAT 146

The average value of a function f, on a ≤ x ≤ b, with f continuous on that interval, is: Wednesday, August 30, 2017 MAT 146

Determine the average value of y = x2 on [0,3]. Wednesday, August 30, 2017 MAT 146

Wednesday, August 30, 2017 MAT 146

Determine the average value of the function here, for the specified interval. Determine a value c such that f(c) generates that average value. Wednesday, August 30, 2017 MAT 146

Wednesday, August 30, 2017 MAT 146