Calculus II (MAT 146) Dr. Day Wednesday, January 24, 2018 Return Quiz #2 Questions: Area Between Curves (Sec 6.1) Another Application of the Definite Integral: Average Value of a Function (6.5) Quiz #3 For Next Time . . . Wednesday, January 24, 2018 MAT 146

Integration Applications: Area Between Curves (6.1) Wednesday, January 24, 2018 MAT 146

Wednesday, January 24, 2018 MAT 146

Area Between Curves Calculate the area between the graphs of y = x2 + 2 and y = 1 – x for 0 ≤ x ≤ 1. Wednesday, January 24, 2018 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, January 24, 2018 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, January 24, 2018 MAT 146

Wednesday, January 24, 2018 MAT 146

Wednesday, January 24, 2018 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, January 24, 2018 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, January 24, 2018 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, January 24, 2018 MAT 146

Area Between Curves Calculate the area between the graphs of y = 2x3 – 1 and y = x – 1 for 1 ≤ x ≤ 2. Wednesday, January 24, 2018 MAT 146

Wednesday, January 24, 2018 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, January 24, 2018 MAT 146

Wednesday, January 24, 2018 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, January 24, 2018 MAT 146

Wednesday, January 24, 2018 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, January 24, 2018 MAT 146

Wednesday, January 24, 2018 MAT 146

Wednesday, January 24, 2018 MAT 146

Average Values Question: What was the average temperature between midnight and noon yesterday? Wednesday, January 24, 2018 MAT 146

If all we know is that it was 25º F. at midnight and that it was 41º F If all we know is that it was 25º F. at midnight and that it was 41º F. at noon, the average temperature is: Wednesday, January 24, 2018 MAT 146

If we also know that it was 36º F If we also know that it was 36º F. at 6 am, we can re-compute an average temperature: Wednesday, January 24, 2018 MAT 146

What if we know hourly readings? 12 midnight 25° F 4 am 33° F 9 am 38° F 1 am 27° F 5 am 34° F 10 am 39° F 2 am 29° F 6 am 36° F 11 am 40° F 3am 31° F 7 am 37° F 12 noon 41° F 8 am Wednesday, January 24, 2018 MAT 146

Wednesday, January 24, 2018 MAT 146

What if we know temperatures every minute? Every second? Wednesday, January 24, 2018 MAT 146

The average value of a function f, on a ≤ x ≤ b, with f continuous on that interval, is: Wednesday, January 24, 2018 MAT 146

Determine the average value of y = x2 on [0,3]. Wednesday, January 24, 2018 MAT 146

Wednesday, January 24, 2018 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, January 24, 2018 MAT 146

Wednesday, January 24, 2018 MAT 146