Applications of the Integral

Problem : Find the average value of f (x) = | sin(x)| from x = 0 to 2Π .

Solution for Problem 1 >>

We compute

	=	| sin(x)| dx
	=	sin(x)dx
	=	(- cos(x)|0 Π )
	=

We could also have observed that in a previous section we found the area under this function to be 4, which is by definition the product of 2Π and the average value.

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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?

Solution for Problem 2 >>

We have

	=	100(1.05)t dt
	=	|0 3
	=	(1.053 - 1)

or approximately 107.69 dollars.

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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) .)

Solution for Problem 3 >>

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	=

which is approximately equal to 0.79 . Notice that this result depends critically on the interpretation of the word "average". Here we are thinking of average of the y -coordinate as a function of x . If instead we considered the y -coordinate as a function of arc length around the circle, we would be asking for the average of sin(t) on the interval [0, Π] , which is 2/Π .

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