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 bounded by the the x axis and the function y = 16 - x^2.

32/3
36
256/3
500/3
972
none of these

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21. 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

The answer is (ln 2)/6.

As by the definition, the average value of f(x) = 1/x on the interval [4, 16] is equal to the definite ...

Solution Summary

The area of a bounded region is found using integrals. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question. Step by step calculations are given.

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

1. The shaded region R, is bounded by the graph of y = x^2 and the line y = 4.
a) Find the area of R.
b) Find the volume of the solid generated by revolving R about the x-axis.
c) There exists a number k, k>4, such that when R is revolved about the line y = k, the resulting solid has the same volume as the solid in par

Let R be the shaded regionbounded by the graphs of y=sqaure root of x, and y=e to the power of -3x, and the vertical line x=1.
a) Find the area R
b) Find the volume of the solid generated when R is revolved about the horizontal line y=1.
c) The region R is the base of a solid. For this solid, each cross section perp

Find the lower sum for the regionbounded by f(x)=9-x^2 and the axis between x=0 and x=3.
9-27/(2n) + 27/(6n^2)
9+27/(2n) + 27/(6n^2)
18+27/(2n)-27/2
18-27/(2n)-27/(6n^2)
none of the above

(1) Find the volume of the solid bounded by the paraboloid x2 + y2 = 2z, the plane
z = 0 and the cylinder x2 + y2 = 9.
(2) Find the volume of the region in the first octant bounded by x + 2y + 3z = 6.
(3) Find the area of the solid that is bounded by the cylinders x2+z2 = r2 and y2+z2 =
r2.
(4) Find the volume enclosed by t

Use double integration in polar coordinates to find the volume of the solid that lies below the given surface and above the plane region R bounded by the given curve.
1. z=x^2+y^2; r=3
Evaluate the given integral by first converting to polar coordinates.
2. ∬_(0,x)^1,1▒〖x^2 dy dx〗
Solve by double i

See the attached file for full description.
26. Evaluate the triple integral, where E is bounded by the planes y = 0, z = 0, x + y = 2 and the cylinder y^2 + z^2 =1 in the first octant.
Find the volume of the given solid
30. Under the surface z = x^2y and above triangle in the xy-plane with vertices (1, 0), (2, 1), and (4