The Fundamental Theorem of Calculus

Expert reviewed • 22 November 2024 • 9 minute read

HSC Maths Advanced Syllabus

establish and use the formula $\int x^n.dx=\frac{1}{n+1}x^{n+1}+C$ for $n\ne 1$
use the formula $\int_a^bf(x)dx=F(b)-F(a)$ $\int_{a}^{b} f (x) d x = F (b) - F (a)$ , where $F(x)$ $F (x)$ is the anti-derivative of $f(x)$ $f (x)$ , to calculate definite integrals
- understand and use the Fundamental Theorem of Calculus, $F'(x)=\frac{d}{dx}[\int_a^xf(t)dt]=f(x)$ and illustrate its proof geometrically

Note:

Video coming soon!

How are Primitives Related to Integration?

As we know primitives are the anti-derivative of a function, which can be found by using integration. As such, there is a simple, general formula that can be used to find the primitive of a function of the form: $x^n$ , where $n\ne1$ .

\int x^n.dx=\frac{1}{n+1}x^{n+1}+C

Where $C$ is equal to some constant. We will touch on the concept of the constant $C$ while integrating, in a later chapter.

What is the Fundamental Theorem of Calculus?

The Fundamental Theorem of Calculus connects the two central operations of calculus: differentiation and integration. It essentially states that a definite integral can be calculated. This is most commonly completed by determining the primitive of a function’s derivative, using the formula above, and substituting limits into the equation.

There are two main parts to this theorem:

If a function $f(x)$ is governed by the limits $[a,b]$ , and it’s anti-derivative is $F(x)$ , then we can apply the following formula to determine the definite integral.

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

This is often used to calculate the area under the curve, created by a function, as your evaluated answer will be a number.

If a function $f(x)$ is has limits with an open interval, for example $[a,x]$ where $a$ is a constant, and it’s antiderivative is $F(x)$ , it is defined by the formula:

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

This part is often referred to as the Second Fundamental Theorem of Calculus. It tells us that if you construct a function $F(x)$ by integrating another function $f(x)$ from one fixed point $a$ , then the integral of $f(x)$ is just the original function. This formula is generally noted as the signed area function, as it is commonly used to determine the areas of a graph that are located in both the negative and positive axis.

Practice Question 1

Evaluate the following expression: $\int_0^{2} x(x+1)dx$

As explained above, we must use the formula related to the first part of the fundamental theorem, as we are dealing with an integral with limits in the form $[a,b]$ .

However, we must first simplify the integral, so we can apply the formula to find a functions primitive.

\int_0^{2}x(x+1)dx=\int_0^{2}(x^2+x)dx

Now we can integrate:

\int_0^{2}(x^2+x)dx =[(\frac{1}{2+1}x^{2+1})+(\frac{1}{1+1}x^{1+1})]^{2}_{0}\\=[\frac{x^3}{3}+\frac{x^2}{2}]^{2}_0

Using the fundamental theorem formula: $\int_a^bf(x)dx=F(b)-F(a)$ we can now evaluate the integral:

We do this by substituting $b=2$ and $a=0$ into the equation we have found.

[\frac{x^3}{3}+\frac{x^2}{2}]^{2}_0=(\frac{2^3}{3}+\frac{2^2}{2})-(\frac{0^3}{3}+\frac{0^2}{2})\\=(\frac{8}{3}+2)-0\\=\frac{14}{3}

Practice Question 2

Integrate the following: $f(x)=\int_0^x3x^2.dx$

As explained above, we must use the signed area formula related to the second part of the fundamental theorem, as we are dealing with an integral with limits in the form $[a,x]$ .

The integral has already been simplified as much as it can.

Now we must integrate by applying the formula to find the primitive:

\int_0^{x}3x^2dx =[\frac{1}{2+1}3x^{2+1}]^{x}_{0}\\= [\frac{3x^3}{3}]_0^x\\=[x^3]_0^x

Using the fundamental theorem formulas: $F(x)=\int_a^xf(t)dt$ and $\int_a^bf(x)dx=F(b)-F(a)$ we can now determine the original function:

F(x)=(x^3)-(0^3)\\=x^3

Up Next

The Definite Integral

Return to Module 4: Integration