AP Calculus AB: 12.3.3 Infinite Limits of Integration, Convergence, and Divergence
This content explains how improper integrals are formally defined by replacing infinite limits or discontinuities with parameters and then taking the limit. It covers three types of improper integrals: integrals with an infinite upper limit, an infinite lower limit, or both limits extending to infinity. In each case, the improper integral is expressed as the limit of a proper integral. For discontinuous functions, the interval is split at the point of discontinuity, and limits are taken from each side to evaluate the integral.
Infinite Limits of Integration, Convergence, and Divergence
Improper integrals can be expressed as the limit of a proper integral as some parameter approaches either infinity or a discontinuity
Key Terms
Infinite Limits of Integration, Convergence, and Divergence
Improper integrals can be expressed as the limit of a proper integral as some parameter approaches either infinity or a discontinuity
note
Formalizing the idea of improper integrals involves
replacing the infinite endpoint with a parameter whose limit approaches either infin...
Which of the following expressions is equivalent to the improper integral
∫∞af(x)dx?
limb→∞∫baf(x)dx
To evaluate the improper integral∫5−51x2dx,at which values of x should you break the integral?
x = 0
Evaluate ∫2−2 dx/x^4
The integral diverges.
Consider the red region under the curve y=f(x) where x→∞. Which of the following expressions correctly describes the limit of the area of the red region?
limb→∞∫baf(x)dx
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| Term | Definition |
|---|---|
Infinite Limits of Integration, Convergence, and Divergence | Improper integrals can be expressed as the limit of a proper integral as some parameter approaches either infinity or a discontinuity |
note |
|
Which of the following expressions is equivalent to the improper integral | limb→∞∫baf(x)dx |
To evaluate the improper integral∫5−51x2dx,at which values of x should you break the integral? | x = 0 |
Evaluate ∫2−2 dx/x^4 | The integral diverges. |
Consider the red region under the curve y=f(x) where x→∞. Which of the following expressions correctly describes the limit of the area of the red region? | limb→∞∫baf(x)dx |
Evaluate ∫∞−∞−11+x2dx | -π |
Evaluate∫1−11√1−x2dx. | π |
Consider the red region under the curvey=f(x) where x→∞. Which of the following statements about the area isnot correct? |
|
To evaluate the improper integral∫π0sec2xdx,at which values of x should you break the integral? | π / 2 |
Evaluate ∫∞0e−xdx. | 1 |