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Calculus/integral and avg mean value

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Expert: - 3/25/2008


Question
Let f be a continuous function that is defined for all real numbers x and that has
the following properties.

(i)
The integral from 3 to 1 of f (x) =5/2

(ii)
The integral from 5 to 1 of f (x) =10

a. Find the average (mean) value of f over the closed interval [1,3].

b. Find the value of

The integral from 5 to 3 of  (2 f (x )+ 6) dx

c. Given that f(x) = ax + b, find the values of a and b


Answer
a) Since this was asked first, all you can do is approximate the average between 1 and 3 by taking the area (that's the integral) and divide it by the width.

b)  The integral of f(x) from 3 down to 5 is ii-i (5 down to 1) -
(3 down to 1).
Multiply this result by 2.  Integrate the constant 6 from 3 to 5.  Add together to get the answer.

c) Integrate ax+b from 3 downto to 1 and set to 5/2, then
5 down to 1 and set to 10.  Solve these two equations for a, b.

Calculus

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