I can understand the process of differentiation and how that's used to find a tangent by using limits, but have absolutely no idea how integration gives the area under a curve Is there a similar, more intuitive explanation for integration?
I know that an integral can be defined as the limit of a Riemann sum, but how do we even know that the limit of the Riemann sum converges to the integral in the first place? I think that I saw an example using the fundamental theorem of calculus, but it was pretty confusing and I don't think I understand it that well
I can understand the process of differentiation and how that's used to find a tangent by using limits, but have absolutely no idea how integration gives the area under a curve Is there a similar, more intuitive explanation for integration?
I know that an integral can be defined as the limit of a Riemann sum, but how do we even know that the limit of the Riemann sum converges to the integral in the first place? I think that I saw an example using the fundamental theorem of calculus, but it was pretty confusing and I don't think I understand it that well
I can understand the process of differentiation and how that's used to find a tangent by using limits, but have absolutely no idea how integration gives the area under a curve Is there a similar, more intuitive explanation for integration?
I know that an integral can be defined as the limit of a Riemann sum, but how do we even know that the limit of the Riemann sum converges to the integral in the first place? I think that I saw an example using the fundamental theorem of calculus, but it was pretty confusing and I don't think I understand it that well
I can understand the process of differentiation and how that's used to find a tangent by using limits, but have absolutely no idea how integration gives the area under a curve Is there a similar, more intuitive explanation for integration?
I know that an integral can be defined as the limit of a Riemann sum, but how do we even know that the limit of the Riemann sum converges to the integral in the first place? I think that I saw an example using the fundamental theorem of calculus, but it was pretty confusing and I don't think I understand it that well
I can understand the process of differentiation and how that's used to find a tangent by using limits, but have absolutely no idea how integration gives the area under a curve Is there a similar, more intuitive explanation for integration?
I know that an integral can be defined as the limit of a Riemann sum, but how do we even know that the limit of the Riemann sum converges to the integral in the first place? I think that I saw an example using the fundamental theorem of calculus, but it was pretty confusing and I don't think I understand it that well
This is a slightly odd mixture of two questions. Your first is looking for intuition, the second is talking about things you learn in a maths degree, and doesn't quite make sense.
To define an integral for some function over some range of x, say from a to b using Riemann sums, you need:
a) a finite sum of rectangles that underestimates its area, the lower sum, and b) a finite sum of rectangles that overestimates its area, the upper sum.
You then take the limit of the upper and lower sums as you let the width of the rectangles to tend to 0, and the number of them to infinity. If both the upper and lower sums tend to the same value, then the function is said to be Riemann integrable, and the common value of the sums is the value of the integral, whose symbol is written ∫abf(x)dx.
So one way to compute the value of an integral is to evaluate the limit of upper and lower sums. However this is very tricky, usually, and you never do it.
Instead, you rely on a result called the Fundamental Theorem of Calculus, that says that if you are interested in the integral of a function f(x), and you can find some other function F(x) that differentiates to f(x), (i.e F′(x)=f(x) as with F(x)=sinx and f(x)=cosx) then
∫abf(x)dx=F(b)−F(a)
To prove this, we have to show that the upper and lower sums of the Riemann integral both converge to F(b)−F(a).
So this theorem turns the process of finding the value of an integral into that of finding an anti-derivative for a function, and evaluating the value of that anti-derivative a couple of times. This is a lot easier than evaluating the limit of an infinite sum.
It's such a shame we don't get taught this sort of stuff. If you can follow the method, most teachers are happy and wont bother explaining why we do things and wont bother with teh real maths behind teh methods.