- #1

noclueatol

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I'm just working through this book for self-study, so I hope this isn't a stupid question.

What values of a and b maximize the value of [itex]\int_{a}^{b}(x-x^{2}){dx}[/itex]?

(Hint: Where is the integrand positive?)

Well, the integrand is positive for 0 < x < 1, and the answer in the back of the book is a=0, b=1. That gives a value of (1/2 - 1/3) = 1/6.

But the problem asks about the expression as a whole, not the integrand. And it doesn't specify that a<b. For large values of x, the integrand will take on a large negative value. Then if you reverse the order of the limits, you get a large positive value. So why wouldn't, say, a=10,000 and b=1 yield a much larger value for the integral than the given answer? Or is it always assumed that a<b?

## Homework Statement

What values of a and b maximize the value of [itex]\int_{a}^{b}(x-x^{2}){dx}[/itex]?

## Homework Equations

(Hint: Where is the integrand positive?)

## The Attempt at a Solution

Well, the integrand is positive for 0 < x < 1, and the answer in the back of the book is a=0, b=1. That gives a value of (1/2 - 1/3) = 1/6.

But the problem asks about the expression as a whole, not the integrand. And it doesn't specify that a<b. For large values of x, the integrand will take on a large negative value. Then if you reverse the order of the limits, you get a large positive value. So why wouldn't, say, a=10,000 and b=1 yield a much larger value for the integral than the given answer? Or is it always assumed that a<b?

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