Blog_Integral_Equations

Tetsuya Matsumoto

Yuze Zhang

Ricky Tindjau

Introduction and Overview:

Students often get confused by the difference between antidifferentiation and integration.

This confusion arises from the power of the Fundamental Theorem of Calculus (FTC).

The FTC says that given a continuous function f(t) there exists a continuous function F(x) such that

F(x) =  Integral _0 ^x {f(t)} dt.

So, in a sense, one can think of antidifferentiation is equivalent to integration.

However, the definitions of these two operations, integration and antidifferentiation are clearly different. The former, (Riemann) integration, is roughly defined as the limit of sum of rectangles under a curve. On the other hand, antidifferentiation is purely defined as the process of finding a function whose derivative is given.

Indeed, integration and antidifferentiation are very close and there is a very deep relationship between differentiation and integration deduced from FTC, which deserves the name “Fundamental Theorem of Calculus”.

Before this theorem is discovered, differentiation and integration was not thought of as inverse operation to one another or not even related.

Typically, students are taught basically these two are the same at high school, if they took a calculus course. This may also be the cause of confusion.

The adoption of analogies is necessary for a better interpretation of the difference between integration and antidifferentiation.

An analogy of antidifferentiation and integration can be rice and wheat, they are both a staple food like how antidifferentiation and integration is the core in integral calculus but are not exactly the same. Rice and wheat are two very similar in composition as both of them are starches, but rice and wheat are very different as well since both of them are used in totally different ways. This is very similar to integration and antidifferentiation as well, both of these are almost identical but in definition are very different and used in different ways.

In addition, one can consider the integral as the shadow of an object under the light and the antiderivative as the shape of the object. The shadow change over time respect to the position of the source of the light. The change of the position cause the variation of the distance between the object and the light source and the variation of the angle of the light. The change of the angle and the distance between the light source and the object lead to the difference of shapes and sizes of the shadow. The size and the shape of the shadow represents the boundary of the integration. The boundary depends on how you use the object and the light. However, the shape of the object is a property of the shadow. The property of the shadow must be similar with the shape of the object because of the transmission of the light (Light transmits in a straight line). The similarity in the shape fixed the property of the shadow’s shape. The property of the shadow represents the antiderivative. The antiderivative is the reverse of the derivative. The reverse of the derivative is fixed over time as a tool for the calculation of the integral.

Ricky Tindjau 45303161

Tetsuya (Tim) Matsumoto 21631163

Yuze (Chronus) Zhang 55456164

Motivation for Integration

Integration is an abstract mathematical operation of “piling up” to represent a shape. For example, we can imagine the Pyramid from ancient Egypt. As we know, the Pyramid is made of many rectangular solids.  Assuming we can only use the rectangular solids to make a Pyramid, one side of the pyramid is like a stair. In order to make the stair smoother so that the pyramid looks more like the perfect square-based-pyramid, smaller blocks can be used. But if we use smaller blocks, the total size of the pyramid also gets smaller, so we need to use more blocks to keep the rough total size. As we use smaller and more blocks, the steps of the stair will be smaller but the number of the stair will increase. By repeating this process, the pyramid can get closer and closer to the perfect square-based-pyramid. The total volume of the pyramid would be closer to the volume of a perfect square-based-planer as well. Our concept of mathematical integration is analogous to this process. Instead of blocks, we just use rectangles to make a stair along a curve smoother. The size of the pyramid could also influence the final volume of the perfect square-based-pyramid. The higher the pyramid is, the more layers of blocks were used to build the pyramid. When all those blocks for building the pyramid are extremely small, in comparison, the total number of the blocks used in a bigger pyramid is larger than a small pyramid. The difference of the total number of blocks used in two different pyramids with different sizes leads to difference on the size of two square-based-pyramids. This is similar to one of the ways to find the area under different curves.

Ricky Tindjau

Yuze (Chronus) Zhang

Tetsuya (Tim) Matsumoto

The first one is rather simple just use the ratio test and prove that the limit of the sequence is <1 for -4<x<4

For part 2 of the question I first identified any similar series, I do this because if u find any similar series using the comparison test is really simple. The harmonic series was really similar to the series and by comparison test the series diverge.

For part 3 of the question, there is (-1)^n in the equation of the series. This normally indicates that using the alternating series test is normally the way to prove for convergence or divergence and it was the case for this question.

All converging series have limits that converge to zero and for diverging series if the limit does not converge to 0 then the series converge.

An exception to this is the harmonic series where the limits converge to 0 but the series diverge.

Example of series

When the limit of the sequence is a finite number when n is sufficiently large then the sequence is converging, if the limit of the sequence is infinity when n is sufficiently large the sequence diverges.

Example of sequence

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