How to tell if an integral is improper?
Understand the Problem
The question is asking for criteria or methods to determine whether an integral is considered improper, which typically involves recognizing infinite limits or integrands that become unbounded.
Answer
An integral is improper if it has infinite limits of integration or if the integrand is unbounded in its range.
Answer for screen readers
An integral is considered improper if it has infinite limits of integration or if the integrand becomes unbounded at any point in the domain.
Steps to Solve
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Identify Infinite Limits
Check if the limits of integration are infinite. An integral is improper if at least one of the limits is infinite, such as in the case of $\int_{a}^{\infty} f(x) , dx$ or $\int_{-\infty}^{b} f(x) , dx$. -
Check for Unbounded Integrands
Examine the integrand for points where it becomes unbounded. This means if there exists a point $c$ in the interval where $f(x)$ approaches infinity, such as in $\int_{a}^{c} f(x) , dx$ where $f(c) = \infty$ or is undefined. -
Combine Both Conditions
An integral is considered improper if either of the above conditions holds true. For example, $\int_{a}^{b} f(x) , dx$ is improper if $b$ is infinite or if $f(x)$ is unbounded at any point in $[a, b]$.
An integral is considered improper if it has infinite limits of integration or if the integrand becomes unbounded at any point in the domain.
More Information
Improper integrals are commonly encountered in calculus, particularly in problems involving limits and areas under curves that approach infinity or have discontinuities.
Tips
- Confusing the conditions for proper and improper integrals. Ensure you take both infinite limits and unbounded behavior of integrands into account.
- Failing to check the entire range of integration for unbounded points; remember to examine all the points in the interval.
