Problem : Find the average value of f (x) = | sin(x)| from x = 0 to 2Π .
We compute
= | | sin(x)| dx | ||
= | sin(x)dx | ||
= | (- cos(x)|_{0} ^{ Π }) | ||
= |
Problem : Suppose Eleanor invests 100 dollars in an account that is compounded continuously with an annual yield of 5 percent, so that the number of dollars in the account after t years is given by A(t) = 100(1.05)^{t} . What is the average amount of money in her account over the first 3 years?
We have
= | 100(1.05)^{t} dt | ||
= | |_{0} ^{3} | ||
= | (1.05^{3} - 1) |
or approximately 107.69 dollars.
Problem : What is the average y -coordinate of a point on the upper half of the unit circle centered at the origin? (You may use that dx = (x/2) + (1/2)sin^{-1}(x) .)
The upper half of the unit circle centered at the origin is the graph of the function f (x) = on the interval [- 1, 1] . The average value of f on this interval equals
dx | |||
= (2) dx | |||
= + sin^{-1}(x)|_{0} ^{1} | = |