Calc: 5.10 Improper Integrals:ChatGPT

What is an improper integral?

An improper integral extends the concept of a definite integral to cases where the interval of integration is infinite or the function has an infinite discontinuity.

Define a Type 1 improper integral.

A Type 1 improper integral involves an infinite interval, such as ∫ₐ^∞ f(x) dx = limₜ→∞ ∫ₐ^ₜ f(x) dx or ∫_{−∞}^b f(x) dx = limₜ→−∞ ∫ₜ^b f(x) dx.

When is a Type 1 improper integral convergent?

It is convergent if the corresponding limit exists as a finite number; otherwise, it is divergent.

Define a Type 2 improper integral.

A Type 2 improper integral occurs when f(x) has an infinite discontinuity at a or b on [a,b]; for example, ∫ₐ^b f(x) dx = limₜ→b⁻ ∫ₐ^ₜ f(x) dx.

What is the geometric interpretation of an improper integral?

If f(x) ≥ 0, the improper integral represents the area under the curve y = f(x) over an infinite region or near a vertical asymptote.

Explain the p-test for ∫₁^∞ 1/xᵖ dx.

The integral converges if p > 1 and diverges if p ≤ 1.

Explain the p-test for ∫₀¹ 1/xᵖ dx.

The integral converges if p < 1 and diverges if p ≥ 1 (since the discontinuity is at x = 0).

What does it mean if an improper integral diverges?

The limit defining the integral does not exist or approaches ±∞, meaning the area under the curve is infinite.

How can l’Hôpital’s Rule help in evaluating improper integrals?

It can be used when the limit of an integrand as x → ∞ or x → a ± is an indeterminate form like 0·∞ or ∞/∞.

What does it mean for an improper integral to converge conditionally?

The integral converges to a finite value, but its absolute value diverges (i.e., ∫|f(x)| diverges).

What is the Comparison Theorem for improper integrals?

If 0 ≤ f(x) ≤ g(x) for x ≥ a and ∫ₐ^∞ g(x) dx converges, then ∫ₐ^∞ f(x) dx also converges; if ∫ₐ^∞ f(x) dx diverges, so does ∫ₐ^∞ g(x) dx.

What is the difference between Type 1 and Type 2 improper integrals?

Type 1 involves an infinite interval (horizontal infinity), while Type 2 involves a discontinuity (vertical infinity).

Give an example of a convergent improper integral.

∫₁^∞ 1/x² dx = 1 (convergent since p = 2 > 1).

Give an example of a divergent improper integral.

∫₁^∞ 1/x dx diverges (logarithmic divergence).

When can an improper integral be split at c ∈ (a,b)?

When f(x) has a discontinuity at c but both ∫ₐ^c f(x) dx and ∫_c^b f(x) dx are convergent, the whole integral is the sum of both.

What is the relationship between improper integrals and probability distributions?

Many continuous probability models (e.g., normal distribution) rely on improper integrals over infinite domains.