And we'll do the intuition for why this happens or why this is true and maybe a proof in later videos. But how would you actually apply this right over here? Well, let's say someone told you that they want to find the derivative. Let me do this in a new color just to show this is an example. Let's say someone wanted to find the derivative with respect to x of the integral from-- I don't know.
I'll pick some random number here. So pi to x -- I'll put something crazy here -- cosine squared of t over the natural log of t minus the square root of t dt. So they want you take the derivative with respect to x of this crazy thing. Remember, this thing in the parentheses is a function of x.
Its value, it's going to have a value that is dependent on x. If you give it a different x, it's going to have a different value. So what's the derivative of this with respect to x? Well, the fundamental theorem of calculus tells us it can be very simple. We essentially-- and you can even pattern match up here.
And we'll get more intuition of why this is true in future videos. But essentially, everywhere where you see this right over here is an f of t. Everywhere you see a t, replace it with an x and it becomes an f of x. So this is going to be equal to cosine squared of x over the natural log of x minus the square root of x.
You take the derivative of the indefinite integral where the upper boundary is x right over here. It just becomes whatever you were taking the integral of, that as a function instead of t, that is now a function x. So it can really simplify sometimes taking a derivative. And sometimes you'll see on exams these trick problems where you had this really hairy thing that you need to take a definite integral of and then take the derivative, and you just have to remember the fundamental theorem of calculus, the thing that ties it all together, connects derivatives and integration, that you can just simplify it by realizing that this is just going to be instead of a function lowercase f of t, it's going to be lowercase f of x.
Let me make it clear. In this example right over here, this right over here was lowercase f of t. And now it became lowercase f of x.
And I will add too the demonstration why the sum of partitions is equal to the whole difference. My analysis professor taught me that it is the Most Important Theorem in Calculus. Add a comment. Michael Hardy Michael Hardy 1. One is what you have written. Ian Ian 90k 2 2 gold badges 71 71 silver badges bronze badges. I would actually argue that the second FTC, or rather its conclusion, is stronger.
I do agree that for purposes of intuition, understanding just the second FTC is sufficient, however. Community Bot 1. The overall idea is fine, of course, just that the detail of justifying that step is tricky to get right. Joe Lamond Joe Lamond 7 7 bronze badges. Upcoming Events. Featured on Meta. Now live: A fully responsive profile. The unofficial elections nomination post. As we learned in indefinite integrals , a primitive of a a function f x is another function whose derivative is f x.
Let's say we have another primitive of f x. Let's call it F x. In indefinite integrals we saw that the difference between two primitives of a function is a constant. That is:. In the first part we used the integral from 0 to x to explain the intuition. However, we could use any number instead of 0.
This does not make any difference because the lower limit does not appear in the result. Here we're getting a formula for calculating definite integrals. Remember that F x is a primitive of f t , and we already know how to find a lot of primitives!
The last step is to specify the value of the constant C. Now, remember that x is a variable, so it can take any valid value. If we make it equal to "a" in the previous equation we get:. We have the value of C. And that's an impressive result. This formula says how we can calculate the area under any given curve, as long as we know how to find the indefinite integral of the function.
And what is F x? It is the indefinite integral of the function we're integrating. We get. The Fundamental Theorem of Calculus, Part 2, is perhaps the most important theorem in calculus. After tireless efforts by mathematicians for approximately years, new techniques emerged that provided scientists with the necessary tools to explain many phenomena.
Using calculus, astronomers could finally determine distances in space and map planetary orbits. Everyday financial problems such as calculating marginal costs or predicting total profit could now be handled with simplicity and accuracy. Engineers could calculate the bending strength of materials or the three-dimensional motion of objects. Our view of the world was forever changed with calculus.
After finding approximate areas by adding the areas of n rectangles, the application of this theorem is straightforward by comparison. It almost seems too simple that the area of an entire curved region can be calculated by just evaluating an antiderivative at the first and last endpoints of an interval. The Fundamental Theorem of Calculus, Part 2 also known as the evaluation theorem states that if we can find an antiderivative for the integrand, then we can evaluate the definite integral by evaluating the antiderivative at the endpoints of the interval and subtracting.
The reason is that, according to the Fundamental Theorem of Calculus, Part 2, any antiderivative works. This always happens when evaluating a definite integral. The region of the area we just calculated is depicted in Figure. Note that the region between the curve and the x-axis is all below the x-axis. Area is always positive, but a definite integral can still produce a negative number a net signed area. For example, if this were a profit function, a negative number indicates the company is operating at a loss over the given interval.
First, eliminate the radical by rewriting the integral using rational exponents. Then, separate the numerator terms by writing each one over the denominator:.
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