Definite Integral
Signed area under a curve.
Key idea
Signed area under a curve.
In depth
∫ₐᵇ f(x) dx is the limit of Riemann sums: the signed area between the graph and the x-axis. Area below the axis counts as negative. It accumulates a quantity over an interval.
Worked example
∫₀² x dx is the area of a triangle with base 2 and height 2, so 2.
Formula
Real-world uses
- Calculating the total distance a car travels from its speed record
- Work done by a force
Common mistake
Treating signed area as total area; area below the axis subtracts.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Definite Integral?
Signed area under a curve. It is a topic at the advanced level in Calculus.
What is the formula for Definite Integral?
The key formula is ∫ₐᵇ f(x) dx.
Where is Definite Integral used in real life?
Real-world examples: Calculating the total distance a car travels from its speed record; Work done by a force.
What should I learn before Definite Integral?
Make sure you are comfortable with Riemann Sums, the previous topic in Calculus.
What is a common mistake with Definite Integral?
Treating signed area as total area; area below the axis subtracts.
How can I practise Definite Integral?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Definite Integral?
Move on to Fundamental Theorem of Calculus I next.
Related advanced topics in Calculus
- Idea of a Limit – The value a function approaches as the input approaches a point.
- One-Sided Limits – Left and right limits must agree for the limit to exist.
- Limit Laws – Limits of sums, products and quotients are computed from the parts.
- Factoring and Rationalizing Limits – Remove 0/0 indeterminacy with algebra.
- Limits at Infinity – End behaviour of functions.
- Infinite Limits and Asymptotes – Function values grow without bound near a vertical asymptote.