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

Solution Preview

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.

1. Determine the value of f(x)=-5x^3+7x on [-1,2].
2. Determine the area of region bound by f(x)=-2x^2+x+6 and the x-axis on [0,3].
3. Determine the area of the regionbounded by f(x)=x^2 - 4 and g(x)=x+2
4. Determine the area of the regionbounded by f(x)=x+6 and g(x)=-x^2+2 on the interval on [-2, 2].
keywords:

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

Please see the attached file for the full problem description
1. Consider the double integral (please see the attached file)
(a) Sketch the region of integration
(b) Write down the integral with reversed order of integration.
2. Find the area of the regionbounded by the following curves (please see the attached file)
S

The figure shows the regionbounded by the x-axis and the graph of. Use Formulas (42) and (43). Which are derived by integration by parts? To find (a) the area of this region; (b) the volume obtained by revolving this region around the y-axis.
Formula (42)
Formula (43).
See the attached files.

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

Question (1)
What is the volume of the solid of revolution obtained by rotating the regionbounded by y = 1 and y = 5 - x^2 around the X-axis.
Question (2)
Find the volume of the solid of revolution obtained by rotating the regionbounded by y = 1 and y = Tan x about the x-axis from x = 0 to x = pi/4.
See attached file

1. Solve the following differential equation:
-2yy' +3x^2 SQRT(4-y^2) =5x^2 SQRT(4-y^2) , -2 < y < +2
2. Let f(x) = ax^2 , a>0 , and g(x) = x^3
Find the value of a which yields an area of PI (i.e. 3.14159) for regionbounded by figure, y-axis and line x=1.

1. Use a double integral to find the area of the regionbounded by the graphs of
y= x^2 and y= 8-x^2. Provide a sketch and use fubini's theorem to determine the order of integration.
2. Determine the best order of integration to find the double integral ??xe^y^2 da Where the region R is in the first quadrant bounded by the

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