The fundamental theorem of calculus is a theorem that links the concept of integrating a function with that of differentiating a function. The fundamental theorem of calculus justifies the procedure by computing the difference between the antiderivative at the upper and lower limits of the integration process. In this article, let us discuss the first, and the second fundamental theorem of calculus, and evaluating the definite integral using the theorems in detail.
Let’s consider a function f in x that is defined in the interval [a, b]. The integral of f(x) between the points a and b i.e. is the area that is bounded by the curve y = f(x) and the lines x = a, x =b and x – axis depicts the area of the region shaded in brown where x is a point lying in the interval [a, b]. Assuming that the values taken by this function are non negative, the following graph depicts f in x.
A(x) is known as the area function which is given as;
Depending upon this, the fundamental theorem of Calculus can be defined as two theorems as stated below:
The first part of the calculus theorem is sometimes called the first fundamental theorem of calculus. It affirms that one of the antiderivatives (may also be called indefinite integral) say F, of some function f, may be obtained as integral of f with a variable bound of integration. From this, we can say that there can be antiderivatives for a continuous function.
Statement: Let f be a continuous function on the closed interval [a, b] and let A(x) be the area function. Then A′(x) = f(x), for all x ∈ [a, b].
Or
Let f be a continuous realvalued function defined on a closed interval [a, b]. Let F be the function defined, for all x in [a, b], by:
Then F is uniformly continuous on [a, b] and differentiable on the open interval (a, b), and F'(x) = f(x) ∀ x ∈(a, b)
Here, the F'(x) is a derivative function of F(x).
The second fundamental theorem of calculus states that, if the function “f” is continuous on the closed interval [a, b], and F is an indefinite integral of a function “f” on [a, b], then the second fundamental theorem of calculus is defined as:
F(b) F(a) = ∫_{a}^{b} f(x) dx
Here R.H.S. of the equation indicates the integral of f(x) with respect to x.
f(x) is the integrand.
dx is the integrating agent.
‘a’ indicates the upper limit of the integral and ‘b’ indicates a lower limit of the integral.
The function of a definite integral has a unique value. The definite integral of a function can be described as a limit of a sum. If there is an antiderivative F of the function in the interval [a, b], then the definite integral of the function is the difference between the values of F, i.e., F(b) – F(a).
Here are the steps for calculating ∫^{b}_{a} f(x)dx
Q.1: Evaluate the integral: ∫_{2}^{3 }y^{2}dy
Solution: Let I = ∫_{2}^{3 }y^{2}dy
As we know,
∫y^{2}dy = y^{3}/3 = F(y)
Therefore, by second fundamental calculus theorem, we know;
I = F(3) – F(2) = 27/3 – 8/3 = 19/3
Q.2: Evaluate the integral: ∫_{1}^{2}[ydy/(y+1)(y+2)]
Solution: By partial fraction we can factorise the term under integral.
y/[(y+1)(y+2)] = [1/(y+1)]+[2/(y+2)] So,
∫y/[(y+1)(y+2)] = logy+1+2logx+2 = F(y)
Hence, by fundamental theorem of calculus part 2, we get;
I = F(2)F(1) = [– log 3 + 2 log 4] – [– log 2 + 2 log 3] I = – 3 log 3 + log 2 + 2 log 4
I = log(32/27)
Get more questions here for practice to understand the concept quickly.
Evaluate using the fundamental theorem of calculus:
What is the first fundamental theorem of calculus?
First fundamental theorem of integral calculus states that “Let f be a continuous function on the closed interval [a, b] and let A (x) be the area function. Then A′(x) = f (x), for all x ∈ [a, b]”.
How many fundamental theorems of calculus are there?
Two basic fundamental theorems have been given in calculus for calculating the area using definite integrals:
First fundamental theorem of integral calculus
Second fundamental theorem of integral calculus
What are the 4 concepts of calculus?
The 4 concepts of calculus are:
Limits and functions
Derivatives
Integrals
Infinite series
What is the second fundamental theorem of calculus?
Second fundamental theorem of integral calculus states that “Let f be a continuous function defined on the closed interval [a, b] and F be an antiderivative of f. Then ∫_{a}^{b} f(x) dx = [F(x)]_{a}^{b} = F(b) – F(a).
Who first proved the fundamental theorem of calculus?
The first published statement and proof of a basic form of the fundamental theorem, strongly geometric, was given by James Gregory. Isaac Barrow proved a more generalized version of the theorem, while his student Isaac Newton finished the development of the enclosing mathematical theory.
204 videos288 docs139 tests


Explore Courses for JEE exam
