LIMITS
IMPROPER INTEGRAL (1 to ∞)
CONVERGENCE
HOW CAN IMPROPER INTEGRALS CONVERGE AT A SPECIFIC VALUE
Improper integrals can converge at a specific value when the function is decreasing towards zero. Both of the examples above diverge because the evaluated integral goes to infinity. If the integral evaluated to a finite number when the limit was taken, they would have converged.
IMPROPER INTEGRAL ACROSS VERTICAL ASYMTOTE
Real-World Applications
WAVES
This example is compelling to me because I grew up in the ocean, and I have always been fascinated by its power and at times perfection. This is a good example of an improper integral that allows you to fill in details that you may not have
A surfer is fascinated by waves, and he wants to know what the best position is on the beach to catch waves. He figures that the wave at his home break is roughly 1 meter, and the wave follows the curve of 1/1+x^2. In order to know the right placement, he needs to know the precise swell size or the area under the curve. How would one find this value?
