2. One example of a real world application using improper integrals would be defining the mass of an object with an infinite density. An example of this would be a black hole. Another example would be in economics where you need to model a situation with an undefined quantity, such as valuations of a company that has unlimited growth potential.

3. The mathematical paradox known as Gabriel's Horn is another puzzling mathematical mystery problems that are presented to us. Take a horn-like object that you can fill up with paint but can never paint it's surface. You also cannot paint it's inner surface despite being able to fill up the horn with paint. It is a two dimensional mathematical shape that has a finite volume but an infinite surface area. Gabriel's Horn can be defined by revolving the graph of the function f(x)= 1/x restricted tox ? 1around the axis by 360 degrees. Finding the volume of the horncan be determined by using the formula below

v = w/abf(a:)2dac. After this you would have to calculate the surface area of the horn itself by using the this formula. 6 S 2 Quit f(a:)x/1 | f'(:z:)2 dz. Assuming the limit exists we can use the improper integral to dene the limit. {ti>3 l3 E:- at\") d1: = lim/ f(a:)d:r:. (1: Unfortunately, when we attempt to calculate the surface area of the horn and try to dene the limit of the function we come to the conclusion that limit doesn't actually exist meaning that Gabriel's Horn has an innite surface area