Definite Integral Calculator
Calculate definite integrals using the Fundamental Theorem of Calculus: ∫[a,b] f(x)dx = F(b) - F(a). Find exact area under polynomial curves.
∫ Calculus📐 ∫[a,b] cxⁿ dx = c·xⁿ⁺¹/(n+1) evaluated a to b🔢 Math
Coefficient c
Exponent n
Lower limit a
Upper limit b
Please enter valid values.
Formula & Reference
| Variable | Symbol | Formula | Units |
|---|---|---|---|
| Definite Integral Calculator | — | ∫[a,b] cxⁿ dx = c·xⁿ⁺¹/(n+1) evaluated a to b | numeric |
Step-by-Step Examples
Example 1
Parabola
∫[1,3] x² dx
- F(x)=x³/3; F(3)=9, F(1)=1/3
- 9-1/3=26/3≈8.667
✓ 26/3 ≈ 8.667
Example 2
Velocity
∫[0,5] 3t² dt
- F(t)=t³; F(5)=125, F(0)=0
- Displacement = 125 m
✓ 125
Example 3
Negative bounds
∫[-2,0] 2x dx
- F(x)=x²; F(0)=0, F(-2)=4
- 0-4=-4
✓ = -4
Real-World Applications
Physics
∫v dt=displacement; ∫F dx=work.
Economics
Consumer surplus from demand curves.
Engineering
Load calculations and signal energy.
Probability
P(a
Common Mistakes to Avoid
⚠️
Forgetting to divide
∫x^n=x^(n+1)/(n+1), not x^(n+1).
⚠️
Wrong order
F(b)-F(a): upper minus lower.
⚠️
n=-1 special case
∫x^(-1)=ln|x|+C, not x⁰/0.
Frequently Asked Questions
What is FTC? ▾
∫[a,b]f=F(b)-F(a) connects derivatives and integrals.
What is an antiderivative? ▾
F where F'=f. Reverse power rule: x^n → x^(n+1)/(n+1).
Definite vs indefinite? ▾
Definite→number; indefinite→function+C.
What does integral represent? ▾
Net signed area between curve and x-axis.
∫e^x? ∫sin(x)? ▾
e^x+C; -cos(x)+C.
Related Math Calculators
Formula Explorer connections
Interpretation: a definite integral accumulates signed area or total change from the lower to upper bound. The Fundamental Theorem evaluates it using an antiderivative. Assumption: the integrand must be integrable on the interval; discontinuities may require an improper-integral treatment.