Improper Integrals
What happens when an integral has an infinite bound or the function has a vertical asymptote? We use limits to define these "improper" integrals and determine if they have finite area.
What Makes an Integral Improper?
An Improper Integral is one where the standard definition of the definite integral doesn't directly apply. This happens in two cases:
Type 1: Infinite Bounds
One or both limits of integration is or .
Type 2: Discontinuity
The integrand has a vertical asymptote within or at the bounds of integration.
The Key Idea: Use Limits!
We replace the "problem" bound with a variable , integrate normally, then take the limit as approaches the problem value.
Converges
Limit exists (finite value)
Diverges
Limit is or DNE
Two Types of Improper Integrals
Type 1: Infinite Bounds
Upper bound is
Lower bound is
Both bounds infinite
Split at any point : evaluate and separately.
Type 2: Discontinuous Integrand
Asymptote at lower bound
Asymptote at upper bound
Asymptote at interior point
Split into two integrals at the asymptote and evaluate both limits.
Important Reference: The p-Integral
Converges if
Diverges if
Worked Examples
Example 1: Infinite Interval (Convergent)
Evaluate .
Even though the region extends infinitely to the right, its total area is exactly 1!
Example 2: Infinite Interval (Divergent)
Evaluate .
The limit is infinite — the area under from 1 to is infinite!
Example 3: Discontinuous Integrand (Convergent)
Evaluate .
There's a vertical asymptote at . The function as .
Even though the function blows up at , the total area is finite!
Practice Quiz
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