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Calculus II (MAT 146) Dr. Day Wednesday January 22, 2014

Calculus II (MAT 146) Dr. Day Wednesday January 22, 2014. Return Quiz #3 6.1 Area Between Curves: Examples, Extensions, Questions 6.5 Average Value of a Function Assignments. Area Between Curves. Calculate the area between the graphs of y = x 2 + 2 and y = 1 – x for 0 ≤ x ≤ 1.

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Calculus II (MAT 146) Dr. Day Wednesday January 22, 2014

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  1. Calculus II (MAT 146)Dr. Day Wednesday January 22, 2014 • Return Quiz #3 • 6.1 Area Between Curves: Examples, Extensions, Questions • 6.5 Average Value of a Function • Assignments MAT 146

  2. MAT 146

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

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  10. Strategies: Area Between Two Curves • 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. MAT 146

  11. Area Between Curves (2) Calculate the first-quadrant area between the graphs of y = √x and y = x2. Show a picture of the enclosed region. (3) 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! (4) 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. MAT 146

  12. Average Values Question: What was the average temperature between midnight and noon yesterday? MAT 146

  13. If all we know is that it was 5º F. at midnight and that it was 21º F. at noon, the average temperature is: MAT 146

  14. If we also know that it was 16º F. at 6 am, we can re-compute an average temperature: MAT 146

  15. What if we know hourly readings? MAT 146

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  17. What if we know temperatures every minute? Every second? MAT 146

  18. The average value of a function f, on a ≤ x ≤ b, with f continuous on that interval, is: MAT 146

  19. Determine the average value of y = x2 on [0,3]. MAT 146

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

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  23. Assignments • WebAssign 6.1 due tonight (1/22) • WebAssign : Calc I Review (T/F) due tomorrow (1/23) • WebAssign6.5 due Friday (1/24) • Also: • Turn in Syllabus Signoff! Quiz #4 due Sunday night: WebAssign (5.5, 6.1, 6.5) MAT 146

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