![]() ![]() The right hand side is just the difference of the values of the antiderivative $F(x)$ at the limits of integration. Why is this a useful theorem? Well, the left hand side is $\int_a^bf(x)dx$, which usually represents the signed area of an irregular shape, which is usually hard to compute. Let $F(x)$ be any antiderivative of $f(x)$. ![]() This theorem relates indefinite integrals from Lesson 1 and definite integrals from earlier in today’s lesson.įundamental Theorem of Calculus Part 2 (FTC 2): Let $f(x)$ be a function which is defined and continuous on the interval $$. This is the fundamental theorem that most students remember because they use it over and over and over and over again in their Calculus II class. Video 5 Fundamental Theorem of Calculus Part 2 (FTC 2) Video 4 below shows a straightforward application of FTC 1. How fast is the area changing? Well, that’s the instantaneous rate of change of $F(x)$…which we know from Calculus I is $F'(x)$…which we know from FTC 1 is just $f(x)$! Examples using FTC 1 The function $F(x)$ represents the shaded area in the graph, which changes as you drag the $b$ slider. In the definition, $F(x)$ is defined as a definite integral, so it represents a signed area, as we learned earlier in today’s lesson. Drag the $b$ slider back and forth to see how the shaded region changes. The variables in the Desmos graph don’t match our notation in the definition above: instead of $t$, Desmos uses $x$ instead of $x$, Desmos uses $b$. Click here to see a Desmos graph of a function $f(x)$ and a shaded region under the graph. Notice that since the variable $x$ is being used as the upper limit of integration, we had to use a different variable of integration, so we chose the variable $t$.Īnother picture is worth another thousand words. This says that $F(x)$ is an antiderivative of $f(x)$! So you can build an antiderivative of $f(x)$ using this definite integral. We’ll start with the fundamental theorem that relates definite integration and differentiation.įundamental Theorem of Calculus Part 1 (FTC 1): Let $f(x)$ be a function which is defined and continuous on the interval $$. Fundamental Theorem of Calculus Part 1 (FTC 1) We’ll follow the numbering of the two theorems in your text. It’s not too important which theorem you think is the first one and which theorem you think is the second one, but it is important for you to remember that there are two theorems. Different textbooks will refer to one or the other theorem as the First Fundamental Theorem or the Second Fundamental Theorem. While most calculus students have heard of the Fundamental Theorem of Calculus, many forget that there are actually two of them. Video 3 The Fundamental Theorems of Calculus For now, we’ll restrict our attention to easier shapes. Since the area enclosed by a circle of radius $r$ is $\pi r^2$, the area of a semicircle of radius $r$ is $\frac.\] Figure 1: A graph of a function f(x) and three shaded regions between it and the x-axis, between x=-2 and x=1įor most irregular shapes, like the ones in Figure 1, we won’t have an easy formula for their areas.
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