Improper Integrals

An improper integral of Type I is an integral whose limits of integration include $\infty$ or $-\infty$ , or both. Remember that $\infty$ is a process (keep going and never stop), not a number! With this in mind, we define

$\int_a^\infty f(x) \,dx= \lim_{t \to \infty} \int_a^t f(x)\, dx,$ and likewise

$\int_{-\infty}^b f(x)\, dx = \lim_{t \to -\infty} \int_t^b f(x)\, dx.$

When an integral runs from $-\infty$ to $\infty$ , we have to break the integral into two pieces:

$\int_{-\infty}^\infty f(x)\, dx = \int_{-\infty}^a f(x) \,dx + \int_a^\infty f(x) \,dx,$ where we can choose any number for the break point

$a$ . (Zero is often convenient.) To evaluate the limits as

$t \to \infty$ or

$t \to -\infty$ , we might need to use L'Hospital's rule.

On Convergence

We'll say that $\displaystyle\int_a^\infty f(x)\,dx$ converges if $\displaystyle\lim_{t\to\infty}\int_a^t f(x)\,dx$ exists, and that the improper integral diverges if the limit doesn't exist.

We define convergence/diverge of $\displaystyle\int_{-\infty}^b f(x)\,dx$ similarly in terms of the existence of its corresponding limit.

For $\displaystyle\int_{-\infty}^\infty f(x)\,dx$ to converge we need both $\displaystyle\lim_{t\to\infty}\int_{a}^t f(x)\,dx$ and $\displaystyle\lim_{t\to-\infty}\int_{t}^a f(x)\,dx$ to exist.

We'll be talking a lot more about convergence and divergence when we get to sequences and series.

The following video explains Type I improper integrals and works out a number of examples.

The Fundamental Theorem of Calculus

The Indefinite Integral and the Net Change

Substitution

Area Between Curves

Volumes

Integration by Parts

Integrals of Trig Functions

Trig Substitutions

Partial Fractions

Strategies of Integration

Improper Integrals

Differential Equations

Models of Growth

Infinite Sequences

Infinite Series

Integral Test

Comparison Tests

Convergence of Series with Negative Terms

The Ratio and Root Tests

Strategies for testing Series

Power Series

Representing Functions as Power Series

Taylor and Maclaurin Series

Applications of Taylor Polynomials

Partial Derivatives

Multiple Integrals

Iterated Integrals over Rectangles

Double Integrals over General Regions

Type I Integrals

On Convergence