Section 8.8 – Improper Integrals. The Fundamental Theorem of Calculus If f is continuous on the interval [ a,b ] and F is any function that satisfies.

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Presentation transcript:

Section 8.8 – Improper Integrals

The Fundamental Theorem of Calculus If f is continuous on the interval [ a,b ] and F is any function that satisfies F '(x) = f(x) throughout this interval then REMEMBER: [a,b] is a closed interval.

Improper Integral Areas of unbounded regions also arise in applications and are represented by improper integrals. Consider the following integral: Intuitively it appears any unbounded region should have infinite area.

Numerical Investigation b Numerically, it appears

Definition: The Integral Diverges

Improper Integral Areas of unbounded regions also arise in applications and are represented by improper integrals. Consider the following integral: Can the unbounded region have finite area?

Numerical Investigation b Numerically, it appears

Definition: The Integral Converges

“Horizontal” Improper Integrals Note: It can be shown the value of c above is unimportant. You can evaluate the integral with any choice. Both integrals must converge for the sum to converge.

Example 1

Improper Integral Technique The technique for evaluating an improper integral “properly” is to evaluate the integral on a bounded closed interval where the function is continuous and the Fundamental Theorem of Calculus applies, then take the offending end of the interval to the limit. On any free-response question, always use the limit notation to evaluate improper integrals. While the statement below may involve less writing, it is mathematically incorrect and will lose you points: DO NOT WRITE THIS!

Example 2 Use L'Hôpital's Rule

Example 2

White Board Challenge Evaluate:

The Fundamental Theorem of Calculus If f is continuous on the interval [ a,b ] and F is any function that satisfies F '(x) = f(x) throughout this interval then REMEMBER: f must be a continuous function.

Improper Integral Areas of unbounded regions also arise in applications and are represented by improper integrals. Consider the following integral: Can the unbounded region have finite area?

Numerical Investigation b Numerically, it appears

“Vertical” Improper Integrals Note: It can be shown the value of c above is unimportant. You can evaluate the integral with any choice. Both integrals must converge for the sum to converge.

Example 1

Example 2 Since the integral is infinite, it diverges (does not exist).

Example 3

White Board Challenge Evaluate:

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