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Section 8.8 – Improper Integrals
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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.
![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.](http://images.slideplayer.com./26/8321121/slides/slide_2.jpg "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.")
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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.
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Numerical Investigation b 1 2 4 6 8 10 20 40 50 100 Numerically, it appears
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Definition: The Integral Diverges
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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?
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Numerical Investigation b 1 2 3 5 8 9 10 20 50 100 Numerically, it appears
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Definition: The Integral Converges
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“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.
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Example 1
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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!
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Example 2 Use L'Hôpital's Rule
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Example 2
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White Board Challenge Evaluate:
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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.
![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.](http://images.slideplayer.com./26/8321121/slides/slide_15.jpg "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.")
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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?
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Numerical Investigation b 0.5 0.4 0.3 0.2 0.1 0.05 0.01 0.001 0.0001 0.00001 Numerically, it appears
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“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.
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Example 1
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Example 2 Since the integral is infinite, it diverges (does not exist).
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Example 3
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White Board Challenge Evaluate:
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