Limits of integration

Definition

Limits of integration are the values that define the interval over which a definite integral is evaluated. They appear as the lower and upper bounds in the integral notation.

5 Must Know Facts For Your Next Test

1. The limits of integration are denoted as $a$ (lower limit) and $b$ (upper limit) in the integral $\int_{a}^{b} f(x) \ dx$.
2. Changing the order of the limits of integration changes the sign of the integral: $\int_{a}^{b} f(x) \ dx = -\int_{b}^{a} f(x) \ dx$.
3. If both limits of integration are equal, then the value of the definite integral is zero: $\int_{a}^{a} f(x) \ dx = 0$.
4. The Fundamental Theorem of Calculus connects differentiation and integration, indicating that if $F'(x) = f(x)$, then $\int_{a}^{b} f(x) \ dx = F(b) - F(a)$.
5. For piecewise functions, it may be necessary to split the integral at points where the function definition changes.

Review Questions

• What happens to a definite integral if you switch its limits of integration?
• How do you evaluate an integral if both limits of integration are equal?
• Explain how to use limits of integration when dealing with piecewise functions.

Related terms

Definite Integral:

A type of integral that calculates the net area under a curve within specified limits, denoted by $\int_{a}^{b} f(x) \ dx$.

Fundamental Theorem of Calculus:

A theorem that links differentiation and integration, stating that if $F'(x)=f(x)$, then $\int_{a}^{b} f(x) \ dx=F(b)-F(a)$.

Piecewise Function: \text{A function defined by multiple sub-functions, each applying to a specific interval within its domain.}