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Suppose that $f$ and $g$ are integrable on $[a, b],$ but neither$f(x) \geq g(x)$ nor $g(x) \geq f(x)$ holds for all $x$ in $[a, b]$[i.e., the curves $y=f(x)$ and $y=g(x)$ are intertwined].(a) What is the geometric significance of the integral$$\int_{a}^{b}[f(x)-g(x)] d x ?$$$$\text { (b) What is the geometric significance of the integral }$$$$\int_{a}^{b}|f(x)-g(x)| d x ?$$
(a) (area above graph of $g$ and below graph of $f$ ) minus (area abovegraph of $f$ and below graph of $g$ )(b) area between graphs of $f$ and $g$
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
Idaho State University
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eso basically, in this both these problems with a, um since the functions air intertwined, they must intersect it at least one point, um, Simon and draw a couple points of intersection here and say, this is a MB. Um, any time you have, um, like, one function being above the other, you either get a positive or a negative area, but then I'm trying to color code it. Then the after they intersect, there's gonna be a portion that crosses it out. They, um, cancel each other out. Um, so what basically happens is part A, since there's no absolute value, Um, there's going to be portions that cancel out. Um, And if you were to actually subtract the values from each other and you drew a new graf, depending, if he didn't like the red before the blue, you know, you'd have an answer that looks something like this. And the way I drew it ended up being like a negative area. But whatever. Um and so you above the X axis and below the X axis would cancel each other out so the areas cancel. So what? What important thing doesn't do when you put the absolute value in there. So let's say it's the same graph. Um, from A to B is drawing a new one for you than everywhere. It dips below the X axis. The absolute value would make it positive, something that, um so in part B, when you get to absolute value now you actually get the area between the curves. And that's the important thing about this problem is that you understand that premise.
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