Please help with the following problem. Provide step by step calculations for each.
The average value of f(x) = 1/x on the interval [4, 16] is
(ln 2)/3
(ln 2)/6
(ln 2)/12
3/2
0
1
none of these
Find the area, in square units, of the region b

The region in the first quadrant bounded by the graphs of y = x and y = x^2/2 is rotated around the line y=x. Find (a) the centroid of the region and (b) the volume of the solid of revolution.

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1.
? Calculate the Taylor Polynomial and the Taylor residual for the function .
? Prove that as , for all .
? Find the Taylor series of f.
? What is the radius of convergence for the Taylor series? Justify your answer.
2.
? Let f:[0,1] be a bo

Sketch the regionbounded by the graph of the functions and find the area of the region
1) f(x) = - x^2+ 4x + 2, g(x) = x + 2
2) f(y) = y(2 - y), g(y) = -y
3) f(x) = 3^x, g(x) = 2x + 1
keywords: integration, integrates, integrals, integrating, double, triple, multiple

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Let S be a bounded nonempty set and let S^2 = {s^2 : s E S}. Show that sup S^2 = max((sup S)^2, (inf S)^2).

Please help with the following problems. Please provide step by step calculations and diagrams.
For the following problem, sketch the region and then find the volume of the solid where the base is the given region and which has the property that each cross-section perpendicular to the x-axis is a semicircle.
1) the regio