1st Fundamental Theorem of Integral Calculus Calculator

The 1st Fundamental Theorem of Integral Calculus establishes a relationship between differentiation and integration. It states that if a function f is continuous on the closed interval [a, b] and F is an antiderivative of f on [a, b], then the definite integral of f from a to b is equal to F(b) - F(a). This theorem connects the concepts of accumulation (integration) and rate of change (differentiation).

What is the 1st Fundamental Theorem of Integral Calculus?

The First Fundamental Theorem of Calculus provides a method to evaluate definite integrals using antiderivatives. It bridges the gap between the two major branches of calculus: differentiation and integration.

Key aspects of the theorem include:

The definite integral of a function f over an interval [a, b] can be found by evaluating its antiderivative F at the endpoints
This connects the concept of accumulation (integration) with the concept of rate of change (differentiation)
The theorem assumes that the function is continuous on the closed interval [a, b]

First Fundamental Theorem of Calculus:

If f is continuous on [a, b] and F is an antiderivative of f on [a, b], then:

∫[a to b] f(x) dx = F(b) - F(a)

How to Calculate Using the Theorem

To apply the First Fundamental Theorem of Calculus:

Identify the function f(x) you want to integrate
Find an antiderivative F(x) of f(x)
Evaluate F at the upper limit (b)
Evaluate F at the lower limit (a)
Subtract the two results: F(b) - F(a)

Note: The antiderivative F(x) is not unique - any antiderivative will work since the difference between any two antiderivatives is a constant.

Worked Example

Let's calculate ∫[1 to 3] 2x dx using the First Fundamental Theorem of Calculus.

Identify f(x) = 2x
Find an antiderivative: F(x) = x² + C (where C is any constant)
Evaluate F(3) = 3² = 9
Evaluate F(1) = 1² = 1
Calculate F(3) - F(1) = 9 - 1 = 8

The definite integral ∫[1 to 3] 2x dx equals 8.

Applications in Calculus

The First Fundamental Theorem of Calculus has numerous applications in calculus and related fields:

Evaluating definite integrals without using Riemann sums
Finding the average value of a function over an interval
Calculating areas under curves
Solving problems in physics and engineering
Understanding the relationship between accumulation and rate of change

Application	Example
Area under curve	∫[0 to 2] x² dx = (2³/3) - (0³/3) = 8/3
Average value	Average of f(x) = x² from 0 to 2 = (8/3)/2 = 4/3
Physics problems	Distance traveled = ∫[t1 to t2] velocity(t) dt

FAQ

What is the difference between the First and Second Fundamental Theorems of Calculus?

The First Fundamental Theorem connects differentiation and integration by showing how to evaluate definite integrals using antiderivatives. The Second Fundamental Theorem establishes that the derivative of an integral function is the original function.

What if the function is not continuous?

The First Fundamental Theorem requires the function to be continuous on the closed interval [a, b]. If the function has discontinuities, the theorem doesn't apply directly, though techniques like limits can sometimes be used.

Can I use any antiderivative?

Yes, you can use any antiderivative since the difference between any two antiderivatives is a constant, which cancels out when you subtract F(b) - F(a).