Application of Integration

Integration is used in just about every field- engineering, finance, physics, statistics, art- you name it. We will cover three applications here: statistics, finance, and physics.

Finance Application

If I contribute the maximum yearly amount to my IRA every year, how much will I have when I am 65 and would like to retire (40 years from now)? The maximum yearly contribution amount to an IRA is $5,000, and the interest rate is about 7% compounding continuously.

We solve this by integrating
Finance application
When I am 65 years old, I will have $1,103,189.06 in my IRA account, that is a lot, but remember that it is in 2053 dollars (inflation will bring the value of a dollar down) and remember that I want to live off my dividends alone. I think I need an extra 3 million in a separate savings account, just in case. How much will I have to contribute to a savings account that earns 3% interest compounding continuously to get to 3,000,000 in 40 years?

We solve this by integrating
Finance application
Integrate the right hand side and then solve for C to get
C = $38,791
. Wow!! This is what I would have to contribute yearly to that savings account to end up with $3,000,000 in my savings account when I retire. (Which is rather unrealistic on a math teacher's salary. I'll just have to stick with the $1,103,189.06 in my IRA when I retire!)


Statistics Applications

Normal curve The Normal Curve is probably the most well-known and useful curves in statistics. Probability is computed by finding the area under the curve for a certain interval -- what we call integration.

Another application of integration to statistics is the ability to compute the average y-value of a function over an interval (not to be confused with the mean or expected value of a PDF function, although integration is used for that as well.)











The lifetime of a battery is a uniform random variable defined on the domain [30,50]. Find the probability that a battery lasts longer than 35 hours.
stats


So, by integrating, we computed that the probability of a battery lasting longer than 35 hours is .75.
(Goldstein, 2014, p.562)


Physics Applications

pva Integration and derivation is used EVERYWHERE in physics. A common use for integration and derivation in calculus and physics is the relationship between the position, velocity, and acceleration of an object in motion.












Another application is computing the work done by a pump:

The rectangular tank shown here, with its top at ground level, is used to catch runoff water. Assume that the water weights 62.4 lb/ft^3.
How much work does it take to empty the tank by pumping the water back to ground level once the tank is full?
Before we find the work done, there are a couple of things we will need to know:
1. Work = Force x Distance
2. Force = Weight
3. Weight = Volume x Weight per unit

physics


(Hass et. al., 2007, p. 434)