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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 .]$

$$\text { If } \quad \int_{a}^{b}[f(x)-g(x)] d x=0$$$$\begin{array}{l}{\text { then the graphs of } y=f(x) \text { and } y=g(x) \text { cross at least }} \\ {\text { once on }[a, b] .}\end{array}$$

True; the integrand must assume both positive and negative values. By the Intermediate-Value Theorem, the integrand must be equal to 0somewhere in $[a, b]$.

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

Campbell University

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University of Michigan - Ann Arbor

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True-False Determine wheth…

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04:03

00:40

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01:05

True or False. Justify you…

01:00

So the best way I could explain this one is Ah, the only way as honest ever, all dissatisfied, everything is satisfied. Um, I mean, if I may have switched everything around, But the only way for these this toe happen is ah for the functions to cross each other at least once. Um, so I know I just said the answer, but if you have a graph of f of X, let's say something like this and it never crosses G of X will say something like this. Then there's definitely an area between them, So this would be, uh, kind of approved by contra positive. So what I'm getting at is wolf. They're the same. Obviously there's no area, but they intersect everywhere. But if you have, let's say something like this and then something like this, the area above this one would have to cancel out the area, this one. But then you could see a okay, Well, they intersected. So that's why you have areas that cancel hum. So hopefully you like my explanation as to why it's true.

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