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True-False Determine whether the statement is true or false. Explain your answer. [In each exercise, assume that $f$and $g$ are distinct continuous functions on $[a, b]$ and that $A$ de-notes the area of the region bounded by the graphs of $y=f(x),$$y=g(x), x=a, \text { and } x=b .]$

$$\text { If } \quad A=\left|\int_{a}^{b}[f(x)-g(x)] d x\right|$$$$\begin{array}{l}{\text { then the graphs of } y=f(x) \text { and } y=g(x) \text { don't cross on }} \\ {[a, b] .}\end{array}$$

$$\text { True }$$

Calculus 1 / AB

Calculus 2 / BC

Chapter 6

APPLICATIONS OF THE DEFINITE INTEGRAL IN GEOMETRY, SCIENCE, AND ENGINEERING

Section 1

Area Between Two Curves

Integrals

Integration

Applications of Integration

Area Between Curves

Volume

Arc Length and Surface Area

University of Michigan - Ann Arbor

University of Nottingham

Boston College

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01:11

In mathematics, integratio…

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01:05

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um, I'm gonna say true in this problem and the reason why music true is the location of the absolute value. Um, because they tell you the area and since they put the absolute value on the outside of the integral from A to B, if I forget what it is, you have X minus f of x dx. Um, then basically, what this boils down to is if this were not the area and they did so I'm gonna prove this kind of a contra positive. If they did intersex somewhere on that interval, then some of the area would cancel out with some of the area over here. Um, so if if that happens, if this happens than the absolute value, would Onley equal this portion this white space that I haven't really covered, um, which the problem tells you that that's not true. The area. So basically, what it boils down to is the only way for this statement to be true is if they don't intersect in there. There's no canceling out, more or less. Um, now, I'm kind of has intend to say, commit completely to true, because are they counting the endpoints? Um, if I can go back and look at it. I feel like it is. It is unclear because they use closed brackets, but I think it's acceptable. And that's where your teacher your book might have. A different answer is if they include A and B, they can intersect it at those end points. And, um, this could still be a true statement. So I'm committing the true, but I'm willing to listen to anybody that says it's false.

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