Fundamental Theorem of Calculus

Here we present two related fundamental theorems involving differentiation and integration, followed by an applet where you can explore what it means.

You can see some background on the Fundamental Theorem of Calculus in the Area Under a Curve and Definite Integral sections.

For this section, we assume that:

Some function f is continuous on a closed interval [a,b]

This means the curve has no gaps within the interval x=a and x=b, and those endpoints are included in the interval.

First Fundamental Theorem

Given the condition mentioned above, consider the function F (upper-case "F") defined as:

F(x) = int_a^xf(t)dt

(Note in the integral we have an upper limit of x, and we are integrating with respect to variable t.)

The first Fundamental Theorem states that:

(1) Function F is also continuous on the closed interval [a,b];

(2) Function F can be differentiated on the open interval (a,b); and

(3) F'(x)=f(x) That is, the derivative of F(x) is f(x).

Second Fundamental Theorem

We continue to assume f is a continuous function on [a,b] and F is an antiderivative of f such that F'(x)=f(x).

The Second Fundamental Theorem of Calculus states that:

int_a^bf(x)dx = F(b) - F(a)

This part of the Fundamental Theorem connects the powerful algebraic result we get from integrating a function with the graphical concept of areas under curves.

To find the area we need between some lower limit x=a and an upper limit x=b, we find the total area under the curve from x=0 to x=b and subtract the part we don't need, the area under the curve from x=0 to x=a.

Fundamental Theorem of Calculus Applet

You can use the following applet to explore the Second Fundamental Theorem of Calculus.

Things to Do

This applet has two functions you can choose from, one linear and one that is a curve. You can:

• Choose either of the functions.
• Drag the (pink) sliders left to right to change the lower and upper limits for our integral.
• Observe the resulting integration calculations on the right.

Choose Function:

Exercises

Find the derivatives:

1. d/dx int_5^x (t^2 + 3t - 4)dt

2. d/dx int_m^x t sin(t^t)dt

3. d/dx int_0^x t sqrt(1+t^3)dt

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Following are some videos that explain integration concepts.

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