Improper Integrals Math Example 4

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Example 4

medium
Evaluate ∫0∞e−x dx\displaystyle\int_0^{\infty} e^{-x}\,dx.

Solution

  1. 1
    lim⁡b→∞[−e−x]0b=lim⁡b→∞(1−e−b)=1\lim_{b\to\infty}[-e^{-x}]_0^b = \lim_{b\to\infty}(1-e^{-b}) = 1.

Answer

11
As b→∞b\to\infty, e−b→0e^{-b}\to0. The total area under exponential decay is finite (equals 1).

About Improper Integrals

Integrals where the interval of integration is infinite (Type I: ∫a∞f(x) dx\int_a^{\infty} f(x)\,dx) or the integrand has an infinite discontinuity on the interval (Type II: ∫abf(x) dx\int_a^b f(x)\,dx where ff blows up at some point in [a,b][a, b]). Evaluated as limits of proper integrals.

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More Improper Integrals Examples

Example 1 easy

Evaluate [formula].

Example 2 hard

Evaluate [formula] (Type II).

Example 3 easy

Does [formula] converge or diverge?

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