Glossary
Convergence
An improper integral **converges** if the limit used to evaluate it results in a finite, real number.
Example:
The improper integral ∫ from 1 to ∞ of 1/x² dx converges to 1, indicating a finite area under the curve.
Definite Integral
An integral with specific upper and lower bounds, which evaluates to a single numerical value representing the net area under a curve over a given interval.
Example:
To find the exact area under f(x) = x² from x=0 to x=2, you would calculate the definite integral ∫ from 0 to 2 of x² dx.
Divergence
An improper integral **diverges** if the limit used to evaluate it approaches infinity, negative infinity, or does not exist.
Example:
The improper integral ∫ from 1 to ∞ of 1/x dx diverges, meaning the area under this curve is infinite.
First Fundamental Theorem of Calculus
This theorem states that if *F* is an antiderivative of *f*, then the definite integral of *f(x)* from *a* to *b* is equal to *F(b) - F(a)*.
Example:
Using the First Fundamental Theorem of Calculus, we can quickly evaluate ∫ from 0 to π of sin(x) dx as (-cos(π)) - (-cos(0)) = 1 - (-1) = 2.
Improper Integral
An integral where at least one limit of integration is infinite, or where the integrand has an infinite discontinuity within the interval of integration.
Example:
Evaluating the area under 1/x² from x=1 to x=∞ requires setting up an improper integral.
Indefinite Integral
An integral without upper and lower bounds, representing the general antiderivative of a function, including an arbitrary constant of integration.
Example:
The indefinite integral of cos(x) is sin(x) + C, where C accounts for any constant term.
Limits
A fundamental concept in calculus used to describe the value that a function or sequence 'approaches' as the input or index approaches some value, often infinity or a point of discontinuity.
Example:
When evaluating an improper integral like ∫ from 0 to ∞ of e^(-x) dx, you must use limits to replace the infinite bound with a variable approaching infinity.