In this section, we will discuss the most important and most used theorem of calculus – the Fundamental Theorem of Calculus (FTC). The Fundamental Theorem of Calculus will help us to:

  1. Provide a way of easily calculating definite integrals.
  2. Establish the connection between derivatives and integrals.

These two parts of the FTC will be discussed in detail.

First Fundamental Theorem of Calculus

Every continuous function has an antiderivative, and every anti-derivative has a derivative. Essentially, as we already established, integration and differentiation are the reverse of each other.

Example 1: Find F'(x) of

Solution:

Find f(v) and v'. Then multiply them by each other.

Since f(t) = t – 1 and v = x,

f(v) = (v) – 1 = v – 1

v' = (x)' = 1

f(v) * v' = (x – 1) * 1 = x – 1

Find f(u) and u'. Then multiply them by each other.

Since f(t) = t – 1 and u = 3

f(u) = (3) – 1 = 2

u' = (3)' = 0

f(u) * u' = (3 – 1) * 0 = (2) * 0 = 0

Example 2: Find F'(x) of

Solution:

Find f(v) and v'. Then multiply them by each other.

Since f(t) = 2t2-1t+2 and v = x2

f(v) = 2(x2)2-1(x2)+2= 2x4-x2+2

v' = (x2)' = 2x

f(v) * v' = (2x4-x2+2)*(2x) = 4x5-2x3+4x

Find f(u) and u'. Then multiply them by each other.

Since f(t) = 2t2-1t+2 and u = x3

f(u) = 2(x3)2-1(x3)+2= 2x6-x3+2

u' = (x3)' = 3x2

f(u) * u' = (2x6-x3+2)*(3x2) = 6x8-3x5+6x2

Example 3: Find F'(x) of

Solution:

Find f(v) and v'. Then multiply them by each other.

Second Fundamental Theorem of Calculus

Essentially, the definite integral of a function f(x) on an interval [a,b] is the net y-units, the change in y, that accumulate between x = a and x = b.

For rate functions r(t), the definite integral on the interval [a,b] is the net y-units, the change in y, that accumulate between t = a and t = b.

This can be denoted as:

The second part of the Fundamental Theorem of Calculus informs us about finding the exact value of definite integrals, that is, finding the area under a curve within an interval.

Guidelines for applying the second FTC:

  1. Integrate the given function to obtain the antiderivative F(x).
  2. Find F(b) and F(a) by substituting b and a into the antiderivative.
  3. Apply the formula:

Example 4: Evaluate the definite integral

Solution:

Integrate the given function to obtain the antiderivative F(x).

Using the constant rule:

Therefore, the definite integral of

Example 5: Evaluate the definite integral

Solution:

Integrate the given function to obtain the antiderivative F(x).

Using the power rule of integration:

Example 6:

Suppose you are given the function

Solution:

Integrate the given function to obtain the antiderivative F(x).

Firstly, apply the integration on the function,

Example 7: Evaluate the integral

Solution:

Integrate the given function to obtain the antiderivative F(x).

First, let's break the integral into 2 separate integrals:

Let's integrate the first integral using the table of integrals. For exponential functions:

Summary of Section

The Fundamental Theorem of Calculus is a theorem that connects differentiation and integration together.

The first part of the theorem states that every continuous function has an antiderivative, and every anti-derivative has a derivative. Integration and differentiation are essentially the opposites of each other. The formula shows that the derivative of an integral of a function is that original function:

The formula can be further expounded by knowing that the formula is an application of the chain rule:

The second part of the theorem states that if we can find an antiderivative for the integrand F, then we can evaluate the definite integral by evaluating the anti-derivative at the endpoints of the interval and subtracting. The calculated difference represents the net change in a quantity.

References:
https://magoosh.com/hs/ap-calculus/2017/ap-calculus-exam-review-fundamental-theorem-calculus/
http://scidiv.bellevuecollege.edu/dh/Calculus_all/Calculus_all.html
http://mathmistakes.info/facts/CalculusFacts/learn/doi/doi.html
http://www.opentextbookstore.com/appcalc/Chapter3-2.pdf