Definite Integral
College/University
Definition
An integral that has start and end values. You can find the definite integral by finding the indefinite integral at the two points specified in the interval and then subtract them. You can then find the area that is between a functions' graph and the x-axis.
Worked examples
\(\int_{1}^{3} 2x\,dx = \left[x^2\right]_{1}^{3} = 3^2 - 1^2 = 8\)
Find the antiderivative, evaluate at the upper limit, then subtract the value at the lower limit.
\(\int_{0}^{\pi} \sin(x)\,dx = \left[-\cos(x)\right]_{0}^{\pi} = -\cos(\pi) - (-\cos(0)) = 2\)
The definite integral gives the net signed area between the curve and the x-axis on the interval.
Common mistakes
- \(\int_{1}^{3} 2x\,dx = x^2\) → \(\int_{1}^{3} 2x\,dx = \left[x^2\right]_{1}^{3} = 8\) You must evaluate the antiderivative at both limits and subtract; don't leave it as a function.
- \(\int_{3}^{1} 2x\,dx = 3^2 - 1^2 = 8\) → \(\int_{3}^{1} 2x\,dx = 1^2 - 3^2 = -8\) Always subtract (lower limit evaluation) from (upper limit evaluation); reversing limits flips the sign.
- \(\int_{a}^{b} f(x)\,dx\) always equals the area under the curve → It gives net signed area; regions below the x-axis count as negative The definite integral sums signed area; use absolute value if you want total area regardless of sign.
Where you'll use it next
Definite integrals are essential for computing areas, volumes of revolution, arc lengths, work, and probability in calculus II and III, differential equations, physics, and statistics.
Found in 1 StudyPug lesson
Definite Integral
UniversityUniversityCalculus 2
Discover what a definite integral means, how it measures area under a curve, and how to evaluate it using the fundamental theorem of calculus.
See also
Reviewed by Pat Cheng, M.Ed. — StudyPug Curriculum Lead · Last updated June 6, 2026