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Writing Suppose that $f$ and $g$ are continuous on $[a, b]$but that the graphs of $y=f(x)$ and $y=g(x)$ cross several times. Describe a step-by-step procedure for determining the area bounded by the graphs of $y=f(x), y=g(x)$$x=a,$ and $x=b$

adding all the values gives the area.

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

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eso There's two ways that you could answer this problem. So imagine you had some functions, and as the direction say, they intersect multiple times. Um so let's say I had, like, a over here is the x value, and And we had be over here and so you can see all these intersection points. Um, so I'm just gonna call them like C one C two C three, and we could have it as many intersection points that you want. So, like, you can just call this CNN or something like that. Um, so basically, what it boils down to is ah, you want to find the integral from A to the first point of intersection of the upper function minus of lower function. Now, here's the thing is, um, we don't know which function is is above the other one, cause it could be f on top or G on on top. Um, because I just drew random graph. Um, so I'm just gonna assume f of X is the higher one now way that you could fix this is you can just do the absolute value because the absolute value would fix that and make it positive and and add to it the in a row from see, want to see to, um and everything else is the same. The in a grand is still the same. Um, otherwise, if they alternate and you might make more sense to switch FX and G of X if you know which ones the upper function. And I'm just gonna write, plus dot, dot, dot Because the pattern continues until you get to that last area from of C sub end to be because you're married, there's their concern. It depends on the problem. How many times they intersect. Ah f of experience. She of x dx, um or this actually works out because if you do the absolute value of f of X, Ministry of Exit actually negates any longer. I guess it undoes does any of the cancelling out So what, You can actually just right if you have a graphing calculator, at least, um, the integral from a to B of f of X minus G of x dx. But I don't think they're looking for that explanation. But if you have a graphing calculator or some computer algebra system, this is a faster way to get your answer.

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