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If the graph of $f(x)$ approaches 0 as $x$ approaches infinity, what does this suggest about the convergence of $\int_a^\infty f(x) dx$ ?

It suggests the integral may converge, but further analysis is needed. The function must approach 0 'fast enough' for convergence.

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If the graph of $f(x)$ approaches 0 as $x$ approaches infinity, what does this suggest about the convergence of $\int_a^\infty f(x) dx$ ?

It suggests the integral may converge, but further analysis is needed. The function must approach 0 'fast enough' for convergence.

If the graph of $f(x)$ oscillates infinitely as $x$ approaches infinity, what does this suggest about the convergence of $\int_a^\infty f(x) dx$ ?

The integral likely diverges, as the area under the curve does not settle to a finite value.

How can you visually identify a discontinuity on a graph that would make an integral improper?

Look for vertical asymptotes or holes within the integration interval.

If the area under the curve of $f(x)$ from $a$ to $b$ is positive, what does this imply about the value of the definite integral $\int_a^b f(x) dx$ ?

The value of the definite integral is positive.

If the area under the curve of $f(x)$ from $a$ to $b$ is negative, what does this imply about the value of the definite integral $\int_a^b f(x) dx$ ?

The value of the definite integral is negative.

How does the shape of the graph of $f(x)$ impact the convergence or divergence of $\int_a^\infty f(x) dx$ ?

If the function decreases rapidly, it is more likely to converge. If it decreases slowly or oscillates, it is more likely to diverge.

What does a graph of $f(x)$ that is always above the x-axis suggest about the integral $\int_a^\infty f(x) dx$ ?

If the integral converges, it will converge to a positive value.

What does a graph of $f(x)$ that is always below the x-axis suggest about the integral $\int_a^\infty f(x) dx$ ?

If the integral converges, it will converge to a negative value.

How can you approximate the value of an improper integral from a graph?

By visually estimating the area under the curve and considering the behavior as x approaches infinity or a point of discontinuity.

If the graph of $f(x)$ has a vertical asymptote at $x=c$ within the interval $[a, b]$ , how does this impact the evaluation of $\int_a^b f(x) dx$ ?

The integral must be split at $x=c$ and evaluated as two separate improper integrals using limits.

What is the difference between evaluating $\int_a^b f(x) dx$ and $\int_a^\infty f(x) dx$ ?

Definite Integral: Direct evaluation using the Fundamental Theorem of Calculus. Improper Integral: Requires expressing as a limit and evaluating the limit.

Compare and contrast the convergence tests for series and improper integrals.

Series: Ratio test, comparison test, etc. Improper Integrals: Direct evaluation via limits, comparison theorems (comparing to known convergent/divergent integrals).

What is the difference between a definite integral and an indefinite integral?

Definite Integral: Has upper and lower bounds, evaluates to a numerical value. Indefinite Integral: Does not have bounds, evaluates to a function + C.

Compare the methods for handling discontinuities within the interval of integration for proper and improper integrals.

Proper Integrals: Discontinuities typically lead to undefined integrals. Improper Integrals: Discontinuities are handled by splitting the integral and using limits.

What is the difference between convergence and divergence?

Convergence: The limit approaches a finite value. Divergence: The limit approaches infinity or does not exist.

Compare the use of u-substitution in proper and improper integrals.

Proper Integrals: u-substitution simplifies the integrand. Improper Integrals: u-substitution simplifies the integrand before taking the limit.

Compare the use of integration by parts in proper and improper integrals.

Proper Integrals: Integration by parts helps to solve the integral. Improper Integrals: Integration by parts helps to solve the integral before taking the limit.

Compare the use of partial fraction decomposition in proper and improper integrals.

Proper Integrals: Partial fraction decomposition helps to solve the integral. Improper Integrals: Partial fraction decomposition helps to solve the integral before taking the limit.

Compare the use of the Fundamental Theorem of Calculus in proper and improper integrals.

Proper Integrals: Fundamental Theorem of Calculus is used to solve the integral. Improper Integrals: Fundamental Theorem of Calculus is used to solve the integral before taking the limit.

What is the difference between the integral of $\frac{1}{x}$ and $\frac{1}{x^2}$ ?

$\int \frac{1}{x} dx = ln|x| + C$ and $\int \frac{1}{x^2} dx = -\frac{1}{x} + C$

What is an improper integral?

An integral where the limits of integration involve infinity or the function becomes unbounded within the interval.

What does it mean for an improper integral to converge?

The limit of the integral approaches a finite value.

What does it mean for an improper integral to diverge?

The limit of the integral approaches infinity or does not exist.

What does 'unbounded' mean in the context of improper integrals?

The function approaches infinity within the integration interval.

What is the role of limits in evaluating improper integrals?

Limits allow us to evaluate integrals with infinite boundaries by approaching infinity.

Define the term 'integrand'.

The function being integrated.

What is a definite integral?

An integral with an upper and lower bound that results in a numerical value.

What is an indefinite integral?

An integral without boundaries.

What is the First Fundamental Theorem of Calculus?

A theorem that allows us to solve definite integrals.

Define the term 'partial fraction'.

Decomposition of a rational function into simpler fractions.