{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

105-5b-ImproperIntegrals1

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

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern