Visualizing the Fundamental Theorem of Calculus

Autor:Greg Petrics

Let's visualize the Fundamental Theorem of Calculus (FTC). Sometimes the simplicity of the mathematical formulation of the FTC hides its subtle complexity, and it's valuable to have a mental image of the relationship between the integral of a function f(x) and its antiderivative F(x). Remember, the FTC says that if F'(x) is equal to f(x), then F(x) can be used to calculate integrals of f(x). Specifically:

This applet illustrates the left and right sides of this equation.

Let's go through the pieces and parts one by one.

On the left is f(x)=3x^2+2, the function to be integrated.
The lower bound, a, is 1 at the outset.
The upper bound, b, is 3 at the outset.
The area under f(x), above the x-axis, and between x=a and x=b is shaded red, and is equal to 30 at the outset. Note: each rectangle in the grid of the graph is 10 units, so you can use those to build intuition that this area is 30.

On the right is F(x)=x^3+2x. Note that F(x) is the the antiderivative of f(x). In other words, F'(x) is equal to f(x) (to check: either use the Monkey Rules, or open Geogebra and type derivative(x^3+2x)).
The difference F(b)-F(a) is shown as a vertical red line segment, and is also equal to 30 at the outset.

There's no obvious reason why the area on the left and the segment on the right should be equal. But this is exactly what the Fundamental Theorem of Calculus affirms: the area on the left and the segment on the right are always equal. Move a and b, being sure that you keep a less than (left of) b, and notice that at ALL TIMES the area on the left, and the segment on the right remain equal. As you do so, the mathematical statement of the FTC dynamically updates showing you the equality. Don't worry about negative integrals in this activity; negative values needlessly complicate visualizing the FTC. To help you get your head around all these moving pieces and parts, here's the same exact applet, but with the function f(x)=cos(x)+2 on the left. Since the derivative of sine is cosine (according to Monkey Rule 5), it must be that the antiderivative of f(x) is F(x)=sin(x)+2x which is on the right. (Go back and review the Monkey Rules if you're totally in the dark about this.)

Again we see that the area on the left is equal to the line segment on the right. The fact that these are equal is exactly what the Fundamental Theorem of Calculus affirms. In the next activity we'll see an intuitive explanation of why the FTC is true.

Visualizing the Fundamental Theorem of Calculus

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