∞
Improper Integrals: Infinite Convergence
AP Calculus BC · Integration Applications
Learning Objectives
Determine conditions for convergence in bounds with an infinite limit.
Compare $p$-series powers for $1/x^p$.
Visualize infinite geometric area collapsing to a finite scalar.
Key Equations
\( \int_{1}^{\infty} \frac{1}{x^p} dx = \lim_{b \to \infty} \int_{1}^{b} x^{-p} dx \)
Evaluates to:
\( = \lim_{b \to \infty} \left[ \frac{x^{1-p}}{1-p} \right]_{1}^{b} \)
Convergence Rules ($p$-Test)
If
$p > 1$
: Converges to $\frac{1}{p-1}$
If
$p \le 1$
: Diverges to $\infty$
Tags
Calculus BC
Improper Integrals
p-Series
Limits
Current Area $\int_1^b$
0.000
State as $b \to \infty$
CONVERGES
Power $p$ in $1/x^p$
2.00
Upper Bound $b$
10
Analytic Solution
\[ \frac{1}{2-1} = 1 \]
Quick Quiz
What happens to the volume if we revolve $y = 1/x$ (which is $p=1$) around the x-axis from $[1, \infty)$?
The volume diverges to infinity.
The volume converges to π. (Gabriel's Horn!)
The volume converges to 1.