Name:

Daniela Yanez 25418161

Tina Sun 58168162

Henry Qiu 50245166

Yifan Jiang 13398169

We always think integral and an antiderivative are the same thing. However, I prefer to say that antiderivative is much more general than integral. Specifically, most of us try to use antiderivative to solve integral problems and just view them as the same thing. In fact, further studies imply that there are many situation that we can not apply antiderivative to solve such problems. For example:. When apply antiderivative, we get the answer (by Wolfram Alpha). As the function is not continuous at , it is only an antiderivative but not a integral.

An antiderivative refers to a different function whose derivative is the original function. What we mean with that is that if we have a function F'(x)=f(x) in a given interval. For example F(x)=, then F’(x)==f(x). But notice that, where C is a constant, can also be the antiderivative of . So us the illustration shows below jellyfish have same function but slightly different sizes, which represents the constants that each family of antiderivative could have.

However, integral can either be the antiderivative of an indefinite integral or an actual number, which means the “anti-derivative” and the “indefinite integral” are the same thing, but the “definite integral” is the area under the curve which is bounded by l and r, which can be actual number.

We all learn about Riemann Sum. We can divide the domain into infinite vertical lines (making infinite subintervals) for Riemann integral in one dimension. We can use rectangular area formula to figure out the value.

So in some general explanation the main differences are:

Antiderivative talks about undoing differentiation, integrals assign number that tell us how big it is.

A definite integral has limits of integration which give us a number as an answer, while an antiderivative give us a function in terms of the independent variables.

It is worth mentioning that there is a Fundamental Theorem of calculus we can use at Riemann integral. It says if G(x) is an antiderivative of g(x). Then we can use G(x) to express the integral of g(x). The Fundamental Theorem of Calculus connect the integral with antiderivative.

But in other dimension( 3-dimension). We can not use this Theorem. We can not find antiderivative of function. At that time, there will be other Theorem we can use———Stokes Theorem and etc.

Name:

Daniela Yanez 25418161

Tina Sun 58168162

Henry Qiu 50245166

Yifan Jiang 13398169

How can we calculate the volume of water in a river or in the ocean? How can we be sure about the surface of a country? Or how can scientist know the area of the plate tectonics?

Not everything in the world has a natural regular shape, an in the most of the cases it is difficult to figure it out what is the best way to calculate the area and volume of this objects. Scientists throughout the years had developed certain ways to make an approach in order to find the area of this regular objects. However it is difficult to say for those objects that do not a have a regular shape.

In order to solve this problem a brilliant scientists, though in dividing this objects in figures that we already know such as the simple square or rectangle. By dividing the given object in pieces it is possible to give an approximation of the area of that object. Where the more pieces we divided in the more accurate the calculation is. So we can say that this method consist in divide the object incalculable times, in figures for which the area is already known.

In secondary school, students calculated the area of plane geometry formed by the straight line and arc. How can we calculate the area formed by random curve? In fact, we can divide the area into several small pieces in order to calculate them one by one. The precondition is that we need to cut the area into the shape, which is easy. When we cut same shape parts, each piece of area can be calculated as a rectangle. And we just need to calculate the sum of small area.

So the most important thing is that the number of segmentation. How to divide the area to make the calculation more accurate? That is, let us calculation is close to the answer. For example, we need to cut a pancake into lots of same shape pieces. We need to cut the pancake as small as possible in order to reduce the waste. That’s the answer. We need to divide the area into incalculable pieces. And we can fit the accurate value. This is the geometric meaning of integration.

As can be seen in the picture below, in order to calculate the area of the horse we can use the area of each square and multiplied them by the number of pieces.