In this video I go over further into improper integrals and this time discuss the type 1, Infinite Intervals, in great detail as well as outlining a definition for it. The first type of improper integrals is when the interval is infinite, such as the integral of a function from x = a to infinity. Even though the interval may be infinite the actual integral may approach a finite number if the limit of the function at positive or negative infinity exists.
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Improper Integrals: Type 1: Infinite Intervals
In my earlier video I went over an Introduction to Improper Integrals and briefly discussed the two types:
Type 1: Infinite Intervals
Type 2: Infinite Discontinuity (or Discontinuous Integrands)
In this video I will go over the Type 1 improper integrals in greater detail as well as outlining the definition.
Type 1: Infinite Intervals
Consider the infinite region S that lies below the curve y = 1/x2, above the x-axis, and to the right of the line x = 1.
You might think that, since S is infinite in extent, its area must be infinite, but let's take a closer look.
The area of the part of S that lies to the left of the line x = t is:
Notice that A(t) < 1 no matter how large t is chosen to be.
We also observe that:
The area of the shaded region approaches 1 as t → ∞, so we say that the area of the infinite region S is equal to 1 and we write:
Using this example as a guide, we define the integral of f (not necessarily a positive function) over an infinite interval as the limit of integrals over finite intervals.
Definition of an Improper Integral of Type 1
In part (c) any real number a can be used.
Any of the improper integrals in the above Definition can be interpreted as an area provided f is a positive function.