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Volume Calculation with Washer Method for a Function h(x) = e^x - ln(x)

This text explains the calculation of volume using the washer method for the function h(x) = e^x - ln(x), as well as finding the absolute maximum and minimum of the function.

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Volume Calculation with Washer Method for a Function h(x) = e^x - ln(x)

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  1. Area = (ex - ln x) dx 1 1/2 y = ex 2 1 -1 1 y = ln x Part (a) = 1.222 or 1.223

  2. 4 3 2 1 1 2 V = p [(4 - ln x)2 - (4 - ex)2] dx 1 1/2 Part (b) To find the volume, we need to use the washer method. R = (4 - ln x)r = (4 - ex ) Volume = 7.515p or 23.609

  3. Part (c) h(x) = ex – ln x The absolute maximum & absolute minimum of h(x) will occur either at an endpoint or where h’(x) = 0.

  4. = 2.342 = 2.718 Part (c) h(x) = ex – ln x Endpoints: h(1/2) = e(1/2) – ln (1/2) h(1) = e(1) – ln (1)

  5. h(.567143) = 2.330 Part (c) h(x) = ex – ln x Critical points: h’(x) = ex – 1/x Graph both functions, then find their intersection point. ex – 1/x = 0

  6. h(1/2) = 2.342 h(.567143) Absolute Min = 2.330 h(1) Absolute Max = 2.718 Part (c)

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