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105-5b-ImproperIntegrals1

# 105-5b-ImproperIntegrals1 - Week 6 Improper Integrals...

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Week 6 Improper Integrals Summary By our definitions, what we will call improper integrals are not really integrals. But we will see below that we can handle them as limits of normal (proper) integrals, and they are not much harder to compute. There are 2 types, which are improper integrals with: Infinite intervals: For example Z 0 e - x dx ; these have an infinite interval of integration, which means that they represent the area of an infinitely long region. But we will see that such a region can still have a finite area, if it is ’thin enough’. There are three types of infinite intervals: those that extend infinitely far to the right ( R a ), to the left ( R a -∞ ), or to both sides ( R -∞ ). Unbounded functions: For example Z 1 0 1 x dx ; these have an integrand that is unbounded somewhere on the interval of integration (in this example at x = 0). So they represent the area of a region that is infinitely tall, which again can have a finite area. Calculating improper integrals Both types can be calculated as a limit of a proper integral. I will illustrate that for both by example: Z 0 e - x dx = lim c →∞ Z c 0 e - x dx = lim c →∞ ( - e - x ) c 0 = lim c →∞ ( - e - c + 1 ) = 0 + 1 = 1 , Z 1 0 1 x dx = lim c 0 + Z 1 c 1 x dx = lim c 0 + ( 2 x ) 1 c = lim c 0 + ( 2 - 2 c ) = 2 - 0 = 2 . So for both types, we circumvent the ’problem’ in the interval using a limit: for infinite intervals we use (leaving out the integrand for brevity): Z a = lim c →∞ Z c a , Z a -∞ = lim c →-∞ Z a c , Z -∞ = Z 0 -∞ + Z 0 = lim c →∞ Z c 0 + lim c →-∞ Z 0 c ; for a point p where the integrand is unbounded we use Z b p = lim c p + Z b c , Z p

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