AcademyIntegral Calculus
Academy
Definite Integrals
Level 1 - Calculus topic page in Integral Calculus.
Principle
A definite integral gives signed accumulation over an interval. Geometrically, it represents signed area between the graph and the \(x\)-axis.
Notation
\(\int_a^b f(x)\,dx\)
definite integral from a to b
\(a,b\)
lower and upper limits of integration
\(F\)
an antiderivative of f
The Core Method
Find an antiderivative and subtract endpoint values.
Evaluation rule
\[\int_a^b f(x)\,dx=F(b)-F(a)\]
Positive area above the axis contributes positively; area below the axis contributes negatively.
Worked Cases
Question
Compute \(\int_1^3 2x\,dx\).
Answer
An antiderivative of \(2x\) is \(x^2\). Evaluate at the endpoints: \(x^2\big|_1^3=3^2-1^2=9-1=8\).
Examples
Question
Compute \(\int_{-1}^{1}x\,dx\).
Answer
An antiderivative is \(x^2/2\). Evaluate: \(\frac{1^2}{2}-\frac{(-1)^2}{2}=\frac12-\frac12=0\). The positive and negative signed areas cancel.