Week #0: Improper Integrals Goals: Improper Integrals
Improper Integrals - Introduction - Improper Integrals So far in our study of integration, we have dealt with functions that were always continuous on the interval that we were integrating on. When we integrated f(x) over [ a, b ], the function f(x) did not have a discontinuity in [ a, b ], and a and b were always finite. In this section we will introduce the concepts of () integrating f(x) over an infinite interval (so somehow infinite horizontally, and (2) integrating f(x) over an interval [ a, b ] where the f(x) contains a discontinuity in [ a, b ] (i.e. somehow infinite vertically.
Improper Integrals - Introduction - 2 Problem. Give an example of each type of improper integral. In what sense are these integrals improper, compared to our earlier proper integrals?
Improper Integrals - Infinite Interval - Improper Integrals of Type : Infinite Intervals These integrals contain and/or in their limits of integration: a f(x) dx or b f(x) dx. For example, consider the integral x 2 dx. By calculating this integral, we will be finding the area under f(x) = x 2 between x = to x = and we will be able to show that its value is finite, even though the interval of integration is infinite.
Improper Integrals - Infinite Interval - 2 x 2 dx. f(x) = x 2...
Problem. Using the rule stated below, calculate justify the statement made on the previous page. Rule: a f(x) dx = lim t ( t a Improper Integrals - Infinite Interval - 3 f(x) dx ) dx, and so x2.
Improper Integrals - Infinite Interval - 4 ( t dx = lim x2 t ) x 2 dx. f(x) = x 2...
Improper Integrals - Infinite Interval - 5 Problem. Is the area under the graph of f(x) = e 3x, extending from x = 0 to infinity, finite? If so, what is its value?
Improper Integrals - Rules for Powers of x A nice way to explore improper integrals of Type I is to ask how the family of functions /x p, for various positive values of p, relate to 2 each other, and what happens to them when we integrate them from to. As you can see, all these functions 0 cross at the point (,). Improper Integrals - Rules for Powers of x - x x x 2 0 2 3
Improper Integrals - Rules for Powers of x - 2 Problem. Which of the following expectations seems reasonable to you? A. B. C. /x p dx is more likely to be finite when p is small, for then the anti-derivative will be smaller. /x p dx is more likely to be finite when p is small, for then the graph of /x p is closer to the horizontal axis. /x p dx is more likely to be finite when p is large, for then the graph of /x p is closer to the horizontal axis. D. /x p dx is more likely to be finite when p is large, for then the graph of /x p is farther from the vertical axis.
Improper Integrals - Rules for Powers of x - 3
Improper Integrals - Rules for Powers of x - 4 Nomenclature An improper integral converges if the limit from the definition approaches a single finite value. An improper integral that does not converge is said to diverge.
Problem. Calculate x p dx Improper Integrals - Rules for Powers of x - 5
Improper Integrals - Rules for Powers of x - 6 x p dx
Identify whether these examples of integrals of the form converge or diverge: Improper Integrals - Rules for Powers of x - 7 a /x p dx x dx x dx x 2 dx
Extension: Doubly Improper Integrals Doubly Improper Integrals - f(x) dx Must be broken up into two single improper integrals. If either integral diverges, then the original integral diverges. To converge, both single improper integrals must converge.
Doubly Improper Integrals - 2 Problem. Find x 2 e x3 dx.
Doubly Improper Integrals - 3 x 2 e x3 dx.
Doubly Improper Integrals - 4 x 2 e x3 dx.
Improper Integrals - Discontinuous Integrand - Improper Integrals of Type 2: Discontinuous Integrand These are integrals, b a f(x) dx, where f has a vertical asymptote somewhere in [ a, b ]. For example, consider the integral x 2 dx. 0 Even though the function f(x) = has a discontinuity in the specified interval (at x = 0), x2 we will learn techniques that will show that this integral diverges to infinity. That is, the area under f(x) = between x = 0 and x = is x2 infinite.. f(x) = x 2
Improper Integrals - Discontinuous Integrand - 2 Like Type problems, the solution technique for. Type 2 problems involves taking a limit. In this case, since the function is undefined at the x f(x) = location of its vertical asymptote, say at x = a, x 2 we replace a with t and then take the limit as t approaches a.
Improper Integrals - Discontinuous Integrand - 3 Problem. Calculate dx and so verify that the area under the 0 x2 graph of from x = 0 to x = is infinite. x2
Problem. Calculate 4 0 x 2 + x 6 dx. Discontinuous Integrand - Examples - You can easily miss seeing that this is an improper integral!
Discontinuous Integrand - Examples - 2 4 0 x 2 + x 6 dx
Improper Integrals - Escape Velocity - Application - Escape Velocity A perfectly reasonable question at this point is When would we ever integrate over an infinite interval? The answer is more often than you might expect! Here is a classic example from astrophysics. The Universal Law of Gravitation gives the force of attraction between two masses m and m 2 (in kilograms) which are a distance of r meters apart by the formula F = Gm m 2 r 2. This formula is especially interesting when m 2 is the mass of the Earth, and m is the mass of an object in its gravitational field.
Improper Integrals - Escape Velocity - 2 Write down the integral that represents the work required to move a mass of m kg from the surface of the earth (r 0 from the center) to a point completely out of the reach of Earth s gravity.
Improper Integrals - Escape Velocity - 3 Evaluate the integral you found, using the following constants: m = 70 kg, the rough mass of a person m 2 = 6 0 24 kg as the mass of the earth G = 6.67 0 N m 2 /kg 2 is the universal gravitation constant r 0 = 6.38 0 6 m is the distance from the center to the Earth s surface
Improper Integrals - Escape Velocity - 4 To escape the Earth s pull without further assistance, an object must be moving fast enough so that its kinetic energy while moving at the Earth s surface is equal to the amount of energy we just found. If kinetic energy of an object moving at velocity v is given by E = 2 mv2, how quickly must an object at the surface of the earth be moving escape the Earth s gravitational pull completely? (This speed is the escape velocity for the Earth.)