Area Function
Let us consider a function f(t) continuous in the interval [a, b], as shown in the below image:
Now, let’s mark some point x between a and b on the graph.
Now, we must find the area under the curve y = f(t) between the interval [a, x].
So, the area under the curve between a and x is the definite integral from a to x of f(t) dt, is
A(x) = ∫ax f(t) dt
Here A(x) is known as the area function and it is helpful in finding the fundamental theorem of calculus.
In other words, A(x), i.e., ∫ax f(t) dt is the area of the region bounded by the curve y = f(t) on the x-axis with coordinates a and b. If x is a point in between [a, b] interval then ∫ax f(t) dt will represent the area of the shaded region, which is A(x).
Example:
Let’s find out the value of A(x) for function y = 2x between x = 2 and x = 6.
A(x) = ∫26 2x dx
⇒ A(x) = [x2]26
⇒ A(x) = 62 – 22
⇒ A(x) = 36 – 4
⇒ A(x) = 32 square units
Learn More: Area Under The Curve
Fundamental Theorem of Calculus | Part 1, Part 2
Fundamental Theorem of Calculus is the basic theorem that is widely used for defining a relation between integrating a function of differentiating a function. The fundamental theorem of calculus is widely useful for solving various differential and integral problems and making the solution easy for students.
This is widely used in the fields of physics, engineering, medicine, economics, biology, space exploration, statistics, pharmacology, and many more.
Before learning about the fundamental theorem of calculus let’s first learn about calculus and others. In this article, we will learn about calculus, area function, the fundamental theorem of calculus, and others.
Table of Content
- What is Calculus?
- Differential Calculus
- Integral Calculus
- Area Function
- Fundamental Theorem of Calculus
- First Fundamental Theorem of Calculus (Part 1)
- First Fundamental Theorem of Calculus Proof
- Finding Derivative using Fundamental Theorem of Calculus
- Second Fundamental Theorem of Calculus (Part 2)
- Remark on Second Fundamental Theorem of Calculus
- Second Fundamental Theorem of Calculus Proof
- Fundamental Theorem of Calculus Examples