In mathematical notation:<\/p>\n\n\n\n But that still leaves us with the question: <\/p>\n\n\n\n Why do we care about area in physics?<\/em><\/strong><\/p>\n\n\n\n I think the answer to that question is that really, we very\nrarely care about the literal \u201carea\u201d as in area of some planar region, measured\nin square meters. But we frequently care\nabout other physical values that we can represent as areas. And the methods we\nuse to find those areas can be extended to other useful things.<\/p>\n\n\n\n To explain this, in this post we will step back and look at\nhow you can find area in math class before<\/em>\nyou learn the Magic Theorem.<\/p>\n\n\n\n [Just for extra emphasis, I will say again: what follows is not a method you will need in AP Physics. In fact, except for a few days, you won\u2019t even use this method in math class either. But if you want to understand what the Magic Theorem does for you, you have to understand what it replaces.]<\/em><\/p>\n\n\n\n In math class one day, your teacher might ask you to find the area between the x-axis and the function f(x)=x2<\/sup><\/em>:<\/p>\n\n\n\n Now maybe rectangle areas are the only ones you know: <\/p>\n\n\n\n area = length times width <\/p>\n\n\n\n You are kind of stuck. But your teacher asks you to keep trying and says they are willing to accept an approximate<\/em> answer. Oooh. Now there is a way forward here.<\/p>\n\n\n\n You could slice the region\ninto a handful of rectangles. Let\u2019s start with five of them:<\/p>\n\n\n\n You could then calculate the\narea of those rectangles and add them up to get a rough idea of the area of the\nentire region. [Notice, I say we COULD\ndo this. It\u2019s a tedious task to do by\nhand, and in the end we won\u2019t have to do it. But we could if we wanted to. As\nyou can see, I let Desmos do the calculations for us.]<\/p>\n\n\n\n In the diagram above, the\nrectangles are INSIDE the region, so we have underestimated the area. We could have\nchosen to put them OUTSIDE, thus overestimating:<\/p>\n\n\n\n So we can see that the area\nwe are looking for is more than 30 and less than 55. That\u2019s an approximation, but it\u2019s better than\nno knowledge at all. It might even\nsatisfy your math teacher for a while. But then they come back to you with a\nrequest: <\/p>\n\n\n\n \u201cI like your method, but I\nwish it were more precise. Can you\ndivide your graph into more, finer intervals? \nLike maybe 10 of them? Or even 50? \n<\/p>\n\n\n\n Well, you can do this too.\nHere are 10 \u201coutside\u201d rectangles:<\/p>\n\n\n\n And here, side by side, we\nhave 50 \u201cinside\u201d rectangles and 50 \u201coutside\u201d:<\/p>\n\n\n\n We can now make two\nobservations about what happens when you use lots of rectangles:<\/p>\n\n\n\n 1. The difference between \u201cinside\u201d\nand \u201coutside\u201d estimates becomes smaller and smaller.<\/p>\n\n\n\n 2. The amount that you are\noverestimating or underestimating by also gets smaller and smaller.<\/p>\n\n\n\n If we could some how let the\nnumber of rectangles approach infinity, we would know the true area of this\nregion. In fact, that limit is the definition<\/em> of the area of this region. <\/p>\n\n\n\n This way of finding area is\ncalled the Method of Reimann Sums. The\narea is often expressed using summation notation:<\/p>\n\n\n\n Setting up these expressions and then calculating the limit as the number of rectangles approaches infinity can be quite daunting. Mastering this method is a rite of passage in calculus class and I admit that it is possible you will not enjoy the process. It is good to understand what Riemann Sums are, but it is also really nice to not have to use them. That\u2019s what is so magical about the Fundamental Theorem — it is so much easier that it feels like magic. Just to remind you of what we do:<\/p>\n\n\n\n 1. We find a function which\nhas x2<\/sup> as its derivative. For\nexample:<\/p>\n\n\n\n 2. According to our magic\ntheorem, the area we want is the change in that antiderivative:<\/p>\n\n\n\n That’s not so bad! But also, now we have all the pieces in place to answer the question that started us down this road: why do we care about area in physics? <\/p>\n\n\n\n Onward to Part IV of our adventure! <\/p>\n\n\n\n Ahe<\/a>ad to Part IV<\/a><\/p>\n\n\n <\/p>\n\n\n Back <\/a>to Part II<\/a><\/p>\n\n\n\n Go to Summer Reading Outline<\/a><\/p>\n\n\n <\/p>","protected":false},"excerpt":{"rendered":" The Magic Theorem, Part III The previous two posts have wrestled with the relationship between an area (the \u201cdefinite integral\u201d) and antiderivative (the \u201cindefinite integral\u201d). In short, they are related by the Fundamental Theorem, or as I have been calling … Continue reading ![\"\"](\"http:\/\/advancedmathyoungstudents.com\/blog\/wp-content\/uploads\/2018\/07\/fundamentalanswer.jpg\")
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