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Partial fraction decomposition is a way of "re-writing" a function that appears to be non-integrable as one that is easily recognized to be integrable. There's really nothing special about partial fraction decomposition, and you definitely don't need to understand it to understand 1-variable Calculus. The special part is the integration that comes afterwards - that's the actual "Calculus" part. The most obvious applications of integration are afforded by The Fundamental Theorem of Calculus, which tells us (amongst other things) that finding areas under or between curves is really just integrating. So one extremely useful application of integration is finding areas under curves, or areas between curves. In other words, if you have a weird 2-dimensional shape that can be bounded by curves, then you can use integration to calculate the area of that shape.
Integration is also used for a variety of applications in chemistry, physics, economics, computer science, and engineering. Differentiation and Integration are two of the most important and useful concepts in all of mathematics.
Unfortunately most algebra and calc classes don't teach about conic sections anymore. For those interested, here's a video showing what Marathon is talking about:
[quote=Weatherby270]Partial fraction decomposition is a way of "re-writing" a function that appears to be non-integrable as one that is easily recognized to be integrable. There's really nothing special about partial fraction decomposition, and you definitely don't need to understand it to understand 1-variable Calculus. The special part is the integration that comes afterwards - that's the actual "Calculus" part. The most obvious applications of integration are afforded by The Fundamental Theorem of Calculus, which tells us (amongst other things) that finding areas under or between curves is really just integrating. So one extremely useful application of integration is finding areas under curves, or areas between curves. In other words, if you have a weird 2-dimensional shape that can be bounded by curves, then you can use integration to calculate the area of that shape.
Integration is also used for a variety of applications in chemistry, physics, economics, computer science, and engineering. Differentiation and Integration are two of the most important and useful concepts in all of mathematics.
Calculus lets you do math with things that are changing. How long does it take to get from A to B if your speed is constant VS. how long does it take to get from A to B if your acceleration is constant. When you accelerate, your speed is changing and the math is harder.
Calculus lets you switch back and forth between first order, second order, or third order functions. position= first order. When your position changes over time that's velocity - second order. When your velocity changes over time that's acceleration - third order.
Calculus lets you do math with one dimensional shapes in two dimensions or two dimensional shapes in three dimensions. Want to know the volume of a donut? A circle (2D) moving through space (3D) in a straight line makes a cylinder - easy. What if it rotates around a point instead of moving in a straight line? That's a donut. Calculus lets you move that circle through space and calculate the volume the circle leaves behind as it moves.
I think it's good to learn stuff - exercises your brain and helps you think in new ways. There's value in learning complicated stuff you won't necessarily use.
An engineer friend of mine used calculus to help me make a dip stick to measure fuel in a 550 gallon cylindrical barrel that was also tilted about 10°. Had been 30 years since my calculus days.
I have two sons,,and both are engineers,and had to take all them fancy math courses.I told them I quit taking math when the teacher said Pie R Squared,,,and I knew very well pies are round.They just say,,,go weed the garden again Dad.