How to Evaluate Definite Integrals Using Fundamental Theorem of Calculus with Solved Examples
According to integration definition in Mathematics to find the whole, we generally add or sum up many parts to find the whole.
We know that Integration is basically a reverse process of differentiation, which is defined as a process where we reduce the functions into smaller parts.
To find the summation under a very large scale the process of integration is used.
We can use calculators for the calculation of small addition problems which is a very easy task to do. We use integration methods to sum up many parts in problems where the limits reach infinity.
Some Elementary Standard Integrals in Integration
Different Types of Integrals in Mathematics
Till now we have learned what Integration is. There are two types of Integrations or integrals in Mathematics
Definite Integral
Indefinite Integral
What is Definite Integral?
A Definite Integral has start and end values.
In simpler words there is an interval [a, b].
A definite integral is an integral that contains both the upper and the lower limits.
Definite Integral is also known as Riemann Integral.
Representation of a Definite Integral -
The variables a and b (called limits, bounds or boundaries) are put at the bottom and top of the S, like this:
In this article we are going to discuss what definite integral is, properties of definite integrals which will help you solve definite integral problems and how to evaluate definite integral examples.
Definition of Definite Integral
The Quantity
It is known as the definite integral of f(x) from limit a to b. In the above given formula, F(a) is known to be the lower limit value of the integral and F(b) is known to be the upper limit value of any integral.
There is also a little bit of terminology that we can get out of the way. The number a at the bottom of the integral sign is called the lower limit and the number b at the top of the integral sign is called the upper limit. Although variable a and variable b were given as an interval the lower limit does not always need to be smaller than the upper limit that is b here. The variables a and b are often known as the interval of integration. Let’s understand the concept in a better way by solving definite integral problems.
Properties of Definite Integrals
Area Above - Area Below
The integral adds the area above the axis but the integral subtracts the area below, to obtain a net value.
Adding of functions
The integral of the functions f and g (f+g) generally equals the integral of function f plus the integral of the function g:
Reversing the Interval
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When we reverse the direction of the interval it gives the negative of the original direction.
Interval of Zero Length
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When the interval of the integral starts and ends at the same place, in simpler words if the limit is same then the result is zero:
Adding Intervals
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We can also add two adjacent intervals together, here’s the formula:
These properties will help you solve definite integral problems and how to evaluate definite integral examples.
Let's evaluate definite integral examples and solve definite integral problems.
Questions to be Solved
Question 1) Solve the following definite integral.
\[\int_{-2}^{3} x^{3} dx\]
Solution)\[\int_{-2}^{3} x^{3} dx\]
\[\int_{-2}^{3} x^{3} dx = [\frac{x^{4}}{4}]_{-2}^{3}\]
= \[\frac{81}{4} - \frac{16}{4}\]
= \[\frac{65}{4}\]
= 16.25
Question 2) Evaluate the integral given below.
\[\int_{0}^{\frac{\pi}{2}} cosx dx\]
Solution) Given, \[\int_{0}^{\frac{\pi}{2}} cosx dx\]
= \[\int_{0}^{\frac{\pi}{2}} cosx dx\]
On evaluating the given question,
= \[sin(\frac{\pi}{2}) - sin(0)\]
We know that the value of sin 0 is equal to zero and the value of sin (\[\frac{\pi}{2}\]) is equal to 1.
Therefore , putting the values ,
= 1- 0
= 1
FAQs on Evaluating Definite Integrals and Their Applications
1. What is a definite integral in calculus?
A definite integral is the limit of a Riemann sum that gives the net area under a curve between two fixed limits. It is written as ∫ab f(x) dx, where a and b are the lower and upper limits of integration. The result is a number (not a function) and represents:
- The signed area between the graph of f(x) and the x-axis.
- Positive area above the x-axis and negative area below it.
- The accumulated change of a quantity over an interval.
2. How do you evaluate a definite integral?
To evaluate a definite integral, find an antiderivative and apply the Fundamental Theorem of Calculus. The steps are:
- Find an antiderivative F(x) of f(x).
- Compute F(b) − F(a).
- Evaluate ∫02 3x² dx
- Antiderivative: F(x) = x³
- Result: 2³ − 0³ = 8
3. What is the Fundamental Theorem of Calculus for definite integrals?
The Fundamental Theorem of Calculus states that if F'(x) = f(x), then ∫ab f(x) dx = F(b) − F(a). This theorem connects differentiation and integration by showing that:
- Integration can be reversed using antiderivatives.
- The definite integral gives the exact accumulated change over [a, b].
4. What is the difference between definite and indefinite integrals?
The key difference is that a definite integral gives a number, while an indefinite integral gives a family of functions. Specifically:
- Indefinite integral: ∫ f(x) dx = F(x) + C (includes constant C).
- Definite integral: ∫ab f(x) dx = F(b) − F(a) (no constant needed).
- Definite integrals have limits; indefinite integrals do not.
5. What does a definite integral represent geometrically?
Geometrically, a definite integral represents the net signed area between a curve and the x-axis from x = a to x = b. This means:
- Area above the x-axis is positive.
- Area below the x-axis is negative.
- If a function crosses the axis, positive and negative areas may cancel.
6. How do you evaluate definite integrals using substitution?
To evaluate a definite integral using substitution, change variables and adjust the limits accordingly. The steps are:
- Let u = g(x).
- Find du and rewrite the integral in terms of u.
- Change the limits from x-values to u-values.
- Integrate and evaluate using new limits.
- ∫01 2x cos(x²) dx
- Let u = x² ⇒ du = 2x dx
- New limits: u = 0 to 1
- Result: sin(u) |01 = sin(1)
7. What are the properties of definite integrals?
The main properties of definite integrals help simplify calculations and split intervals. Important properties include:
- ∫aa f(x) dx = 0
- ∫ab f(x) dx = −∫ba f(x) dx
- ∫ab [f(x) + g(x)] dx = ∫ab f(x) dx + ∫ab g(x) dx
- ∫ac f(x) dx + ∫cb f(x) dx = ∫ab f(x) dx
8. Can a definite integral be negative?
Yes, a definite integral can be negative if the function lies mostly below the x-axis on the interval. Since definite integrals measure signed area:
- Area above the axis contributes positively.
- Area below the axis contributes negatively.
9. How do you evaluate definite integrals with trigonometric functions?
To evaluate definite integrals with trigonometric functions, use standard antiderivatives and apply the limits. Common formulas include:
- ∫ sin x dx = −cos x
- ∫ cos x dx = sin x
- ∫0π sin x dx
- Antiderivative: −cos x
- Result: [−cos π] − [−cos 0] = 1 − (−1) = 2
10. What are common mistakes when evaluating definite integrals?
Common mistakes when evaluating definite integrals usually involve limits and signs. Typical errors include:
- Forgetting to substitute both limits into F(b) − F(a).
- Dropping brackets and miscalculating signs.
- Not adjusting limits during substitution.
- Confusing definite integrals with indefinite integrals.
