Innovating, Ironic, Incredible, Integrals

Hello and welcome to my final action project for my Intro to Calculus class. This final unit was based around Integrals and understanding how they can be used in different ways. Integrals are similar to derivatives and can also be used to find the area of a function. One of the other things we used to find the area of a function was looking at Riemann sums. This is the process of drawing rectangles that align with the right, left, or middle of a function to find its area. Once learning how to use riemann sums, we learned how to integrate certain functions and apply the anti-derivative. For example, understanding how to find the antiderivative, and using that to find the area under a function. This all led to this final project which asked us to apply our knowledge using certain numbers given to us. 

"Fundamental theorem of calculus (animation)" by Кубриков is licensed under CC BY-SA 4.0.

Here are my numbers below and the process I took to solve each equation

Numbers: -9 and 2

Equation: f(x) =  -9 * 2x^2

Constant - The value with no change: -9

Coefficient - Placed before multiplying a variable: 2

Exponent - The number that shows how much a variable is changing: x^2

Here’s my equation visualized using Desmos:

“Equation on Display”,RBL, 2023

Rectangle area: L x W

Rectangle 1 area: 0.1 x -0.0018 = -0.00018

Rectangle 2 area: 0.1 x -0.72 = -0.072

Rectangle 3 area: 0.1 x -1.62 = -0.162

Rectangle 4 area: 0.1 x -2.88 = -0.288

Rectangle 5 area: 0.1 x -4.5 = -0.45

Total area:  -0.00018 + -0.072 + -0.162 + -0.288 + -0.45 = -0.97218

Riemann sums are the process that I went through above. This shows that I calculated a small part of the function and made each into rectangles. They can be analyzed from the right, middle, or left of a function and can even be shown with trapezoids. From there, it shows my process to calculate the area from each part of my function. I decided to make sure each rectangle was on a smaller scale because it would make calculating the area a simpler process.

Reverse Power Rule: f(x) = -9 * 2x^2 --> (Multiply -9 * 2) -> -18x^2 

F(x) = ax^b+1/b+1

-18x^3/3 -> -6x^3

Reverse power rule shows how a function can be changed in one way or another. The process is similar to the power rule but can be seen in different ways. Not only this, but the rule can be applied differently depending on the set of numbers used.

 (Integral): 

0.5 : -6(0.5)^3 = -0.25
0 : -6(0)^3 = 0

-0.25 - 0 = -0.25

The -25  is the integral made from putting each side of a function into the derivative and finding a result. Once evaluating it on both sides, they had to be subtracted by each other to get a final answer. In comparison to the -0.97, it shows the change in how a function can change depending on applying the derivative or anti derivative. Each can be used in different ways and might show a different answer. 

This action project gave me an interesting look into how calculus can be used in different ways. I'd say that calculus is understanding how numbers can change over a given period of time and why. These changes can be measured over certain periods of time and depending on the numbers used it might lead to different results. Using my function for example, I used integration to calculate how to find the area under a specific function. Once I figured that out, it help me to understand that integrals and derivatives are both opposite to one another. Not only this, but both are fundamental to understanding calculus. When I figured out how to get the derivative, it wasn't difficult for me to figure out how to get the anti-derivative. This can't just applied to my equation, but to any equation or derivative that deals with these sets of skills.  gave me an outlook on how functions can change and why. It was challenging for me to figure out the integral though I did have the numbers I just needed the right equation. If I were to do this again I might want to see how using different numbers might affect my function.