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The Fundamental Theorem of Calculus

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Instructor: Sarah Wright
The fundamental theorem of calculus is one of the most important equations in math. In this lesson we start to explore what the ubiquitous FTOC means as we careen down the road at 30 mph.

Fundamental Theorem of Calculus

The fundamental theorem of calculus
Fundamental Theorem

The fundamental theorem of calculus says that if f(x) is continuous between a and b, the integral from x=a to x=b of f(x)dx is equal to F(b) - F(a), where the derivative of F with respect to x is equal to f(x). The big F is what's called an anti-derivative of little f. This is one of the most key points in all of mathematics, and it's called the fundamental theorem of calculus. But what the heck does 'fundamental' mean?

Let's look at a graph of velocity as a function of time - so this is f(t) - between some point in time a, and some point in time b. Let's say that f(t) is constant between a and b, just to make things simple. I know that, according to the fundamental theorem of calculus, the integral from a to b of' f(t)dt - so that's the area under this curve - is equal to F(b) - F(a), where F`(t)=f(t). That's the fundamental theorem of calculus. But let's look at it.

Fundamental Theorem in Practice

F(t) is equal to f and the derivative of 30t is equal to 30
Fundamental Theorem Example Graph

Let's say f(t)=30 miles per hour (mph). Let's first find an F that you can take the derivative of to get 30. So here I want to find what's called an anti-derivative of f. Let's say that F=30t. If I take the derivative of 30t, I get d/dt(30t), which is equal to 30d/dt(t), which is just 30. So right now, I know that F`(t)=f and the derivative of 30t is equal to 30, so the derivative of this is equal to this.

Okay, so now I've got f(t), which is 30, and my anti-derivative here is 30t, so let's plug those in. Let's say I've got my velocity here, 30, and I'm integrating it between a and b. According to the fundamental theorem, this is equal to F(b), so that's F, which is 30t, evaluated at t=b, so that equals 30b - F(a), which is 30a. So according to the fundamental theorem of calculus, the integral from a to b of 30dt=30b - 30a. I can simplify this right-hand side to equal 30(b - a). Well let's take a look at our graph. Our graph is just a straight line, so the integral in this case is just this rectangle here. Well, the area of a rectangle is the height times the width. My height is 30, because f(t)=30. My width is b - a. So sure enough, that's what this right-hand side equals: it equals my height times my width, which is exactly the area. So this kind of works, but what does it mean?

f(t)delta t is a Riemann rectangle
Riemann Rectangle

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