Integreality: Integral Calculus in Reality

Integral Calculus in Reality Start INTEGREALITY ByTeam B

Integral Calculus in Reality Next About Integreality Intergreality is a project whose focal point is two significant topics in Integral Calculus which are work and moment of inertia. These topics will be discussed in a broader perspective and in a detailed aspect.

Integral Calculus in Reality Next Meet the Team Hanika Balangue Matt Alexander Cervantes Alexandra Espiritu CEE 11 ESE 11 IEE 11

Integral Calculus in Reality Next Meet the Team Francis Sal Fabiculanan Vinz Andrei Imperial Stanlee Hayes Inosanto CEE 12 CPE 11 CPE 12

Integral Calculus in Reality Next Topic: Work A Definite Integration of force to acquire the exact limit and summation will be presented as a specific topic under this application.

Integral Calculus in Reality Next Introduction

Integral Calculus in Reality Next Work Definition Work is a physics term that describes the energy transfer that occurs when an item is moved across a distance by an external force that is applied in at least part of the displacement direction. Formula Work may be calculated by multiplying the length of the path by the component of the force operating along the path if the force is constant. The work (W) is equal to the force (f) times the distance (d), or W = fd, to represent this notion numerically.

Integral Calculus in Reality Next Problem and Solution

Integral Calculus in Reality Next Problem Ariel has been working in Onel’s Mining Company for a decade. He is in charge of supervising their workplace through ensuring safety and organization. Ariel has always been curious on how much work is done every time a cable lifts tons of coal from the mine shaft. Help Ariel find how much work is done when a cable that weighs 6 lb/ft is used to lift 600 lb of coal up a mine shaft 400 ft deep.

Integral Calculus in Reality Next Solution Given: Consider a section of the cable having length x Weight of the cable = 6lb./ft. feet. Then the total weight of this cable plus the Weight of the coal = 600 lb. 900 lbs of coal at its end will be given by: Depth of the mine shaft = 400 ft. Weight/Force = (600 + 6x) lbs. (1)

Integral Calculus in Reality Next Then the work done in lifting this section of the cable through a distance ∆x will be given as: Work = Weight/Force x Distance = (600 + 6x) ∆x lb. – ft. (2) Since the length of the mine shaft through which the cable with the coal needs to be pulled up is 400 feet, we use (2) to calculate the total work done in pulling the entire cable with the 600 lbs. of coal at its end through the mine shaft

Integral Calculus in Reality Next Answer: The total work done in pulling all the coal up the mine shaft is 720,000 lb-ft.

Integral Calculus in Reality Next Real - Life Application

Integral Calculus in Reality Next Upon viewing the world from a broader perspective, it is evident that mathematics does indeed fulfill a significant role in life. To further elaborate, this topic is unconsciously exploited in various aspects, yet with enough knowledge and observation, it will be apparent. Work is known to be the energy transferred to or from a mass or an object via the application of force along with a displacement. Through its simple definition, it can already convey numerous applications and in this section, some examples will be tackled.

Integral Calculus in Reality Next For instance, you just woke up and were looking for a light breakfast. You saw a banana on your dining table and by then you have decided to eat it as a meal. The moment that you pick the banana, is also the moment you exerted work. Why? Because you applied force to move or transfer the position of the banana. To further elaborate, if the banana weighs 1 Newton and if it is lifted 1 Meter above the table, then the work done is 1 Newton Meter, it is obtained through the work formula in constant force which is: W = F × d.

Integral Calculus in Reality Next Another example, imagine having a vacation and your friends decided to do rock climbing. You agreed and proceeded to climb your way up. After successfully reaching the top of the mountain, you needed to pull the rope that you used that is still hanging in a cliff face. It is evident that work is exerted the moment you pull the role upwards to completely reach your position on the top of the cliff. However, The rope becomes lighter as more is pulled in, requiring less force and hence the climber performs less work. Therefore, in this situation, work as an integral or variable is used derived by the formula: W= ∫ baF(x)dx.

Integral Calculus in Reality Next To conclude, this occurrence is normal in our everyday life, thus, work is also applied by us. In everything that we do, even the simplest task namely pushing a cart inside the aisle of a grocery store, lifting your backpack full of stuff on your shoulder, kicking a ball towards the goal when playing soccer, pushing a sliding window sidewards to obtain fresh air and a lot more daily activities that requires force or weigh to have displacement. This realization is significant as this knowledge can be considered to have broad, diverse, yet beneficial applications in life partcularly is relevant to our chosen path or industry which is engineering.

Integral Calculus in Reality Next Topic: Moment of Inertia An Integration on the rotation of the mass to measure its tendency will be presented as a specific topic under this application.

Integral Calculus in Reality Next Introduction

Integral Calculus in Reality Next Moment of Inertia Definition Formula Moment of Inertia is a The Moment of Inertia is mass measurement of an object’s times the square of the resisting angular or rotational perpendicular distance to the acceleration. It is mostly rotation axis: I = mr2. Integrals specified to an axis of rotation as can be useful in calculating the it depends on mass and torque Moment of Inertia especially in around the axis of rotation. deriving its general formula.

Integral Calculus in Reality Next Problem and Solution

Integral Calculus in Reality Next Problem: What is the moment of Inertia of the area in the first quadrant bounded by the curves y2= 6x, x axis and the line x= 2 with respect to the x-axis?

Integral Calculus in Reality Next Solution: First to make it simpler and organized we would need to graph the given functions.

Integral Calculus in Reality Next Now the function y2= 6x (represented as red) gives us a parabola , and the line given by x=2 is represented by the yellow, and orange represents the strip in which dy is the thickness, and its length would be represented as xR - xL. (note: the strip is horizontal because we are solving our answer with respect to the x-axis.) Now to begin it is important to note that moment of inertia is always positive. And since the question states that it would be at the first quadrant we will be getting our answer from the upper right section of the coordinate plane.

Integral Calculus in Reality Next First it is important to remember the formula; Ix= ∫y2dA Now to get the formula needed we must first replace the dA with the given data from the strip, which is dA=∫(xR - xL)dy Now we just replace the dA with the formula we made to represent the strip, in which the xR is the given value of x which is 2, and xL since it touched the parabola will be x= y2 / 6 derived from y2= 6x Ix= ∫y2(xR - xL)dy Ix= ∫y2(2 -y2/6)dy

Integral Calculus in Reality Next To get the limit we y2 = 6x y2 = 12 just have to solve for y2 =6 (2) y= 2√3 y2 in which x would be replaced by 2 : Ix= ∫02sqrt3y2(2 -y2/6)dy Ix= ∫02sqrt3(2y2 -y4/6)dy Overall we would have the equation: And our answer would be: Ix= 11.07

Integral Calculus in Reality Next Real - Life Application

Integral Calculus in Reality Next We are taught the principle of inertia at a young age. We are all aware that it takes force to start something moving, change its direction, or bring it to a complete stop. Our intuitive understanding of how inertia works allow us to exert a certain amount of control over the environment around us. Learning to drive teaches you even more valuable lessons. As I continued to research the moment of inertia, I discovered that an automobile contains numerous components that can be used to demonstrate the moment of inertia, one of which is the flywheel.

Integral Calculus in Reality Next A flywheel is a mechanical device that stores energy as rotational momentum. Torque can be applied to a flywheel to make it spin and thus increase its rotational momentum. This stored momentum can then be applied to any rotating object, most commonly machinery or motor vehicles. In the case of motor vehicles and other moving objects, the rotational inertia of the flywheel can have an effect as a result of gyroscopic motion, preventing the vehicle from changing its direction. Turning and stopping the vehicle becomes difficult if the mass of the flywheel is significant in comparison to the overall mass of the vehicle. Because of this, careful design of a flywheel is required for use in moving vehicles.

Integral Calculus in Reality Next Flywheels are frequently used to provide consistent energy when the primary energy source is intermittent. A flywheel, for example, can be connected to an engine's crankshaft (assuming a manual transmission), storing rotational energy while torque is applied. When the torque is removed, the flywheel can continue to apply torque to the driveshaft, resulting in more consistent power output from the engine. This flywheel configuration was frequently employed on older internal combustion engines, which frequently had engine knock and inconsistent output. Small automobiles may be jerky at low speeds due to their lighter flywheels.

Integral Calculus in Reality Next Insights Hanika Balangue CEE 11

Integral Calculus in Reality Next Insights Matt Alexander Cervantes ESE 11

Integral Calculus in Reality Next Insights Alexandra Espiritu IEE 11

Integral Calculus in Reality Next Insights Francis Sal Fabiculanan CEE 12

Integral Calculus in Reality Next Insights Vinz Andrei Imperial CPE 11

Integral Calculus in Reality Next Insights Stanlee Hayes Inosanto CPE 12

Integral Calculus in Reality Next References 6.4 work. 6.4 Work‣ Chapter 6 Applications of Integration ‣ Part Calculus I. (n.d.). https://sites.und.edu/timothy.prescott/apex/web/apex.Ch6.S4.html. Boundless. (n.d.). Boundless calculus. Lumen. https://courses.lumenlearning.com/boundless-calculus/chapter/applications-of- integration/. Bourne, M. (n.d.). 7. work by a Variable force using integration. intmathcom RSS. https://www.intmath.com/applications-integration/7-work-variable-force.php. Definition and mathematics of work. The Physics Classroom. (n.d.). https://www.physicsclassroom.com/class/energy/Lesson-1/Definition-and-Mathematics-of- Work#:~:text=There%20are%20several%20good%20examples,above%20his%20head%2C%2 0an%20Olympian. Encyclopædia Britannica, inc. (n.d.). Moment of inertia. Encyclopædia Britannica. https://www.britannica.com/science/moment-of-inertia.

Integral Calculus in Reality Next References Energy Education, inc. (n.d) Flywheel, Applications Flywheel - Energy Education Libretexts. (2020, December 21). 6: Applications of integration. Mathematics LibreTexts. https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Calculus_(OpenStax)/06%3A_A pplications_of_Integration. Math24 (n.d.) Calculus: Applications of Integrals https://www.math24.net/moment-inertia Moment of inertia. Math24. (2021, March 5). https://www.math24.net/moment-inertia. Moment of inertia by integration. moment of inertia by integration | MATHalino reviewers tagged with moment of inertia by integration. (n.d.). https://mathalino.com/tag/reviewer/moment-inertia-integration.

Integral Calculus in Reality Next References Openstax. (n.d.) 5.6 Calculating Centers of Mass and Moments of Inertia https://openstax.org/books/calculus-volume-3/pages/5-6-calculating-centers-of-mass-and- moments-of-inertia Study.com. (n.d). Calculating Work Done Using a Definite Integral https://study.com/academy/answer/a-cable-that-weighs-4-lb-ft-is-used-to-lift-900-lb-of- coal-up-a-mine-shaft-650-ft-deep-find-the-work-done-show-how-to-approximate-the- required-work-by-a-riemann-sum-let-x-be-the-distance-in-feet-below-the-top-of-the- shaft-enter-x-i-as-x-i.html Toppr Answr. (n.d). Real life examples of moment of inertia - Definition Moment of Inertia | Definition, Examples, Diagrams (toppr.com) Work. (n.d.). http://electron6.phys.utk.edu/phys135core/modules/m6/work.html. Work done by a variable force. Work as an integral. (n.d.). http://hyperphysics.phy- astr.gsu.edu/hbase/wint.html.

Integral Calculus in Reality Next Conclusion To conclude, the objective of the team, which is to present the two chosen topics and convey the acquired knowledge upon learning the stated topics, was successfully achieved by each member. This claim can be validated by this recorded video presentation wherein my team delivered our content in this project.

Integral Calculus in Reality Start THANK YOU Signing Off