Question

can someone help me visualize something about integration?

Answer 1

I tend to see integration like multiplication from a logical perspective, and even from how Riemann sums are defined, since both can be defined as a repetition of addition in a sense.
I see integration as the function-level of multiplication, basically. The tool that can multiply even patterns, or find the area of irregular shapes.
\[Displacement(t) = \int\limits\limits_{?}^{?}Velocity(t)dt, Displacement = Velocity * time\]
It can be extended to other shapes and coordinate systems, too.
\[Volume(x) = \pi \int\limits_{?}^{?}Radius(x)^2dx, V = pir^2h\]

Answer 2

so umm, can someone help me with the formula of calculating arclength. It makes sense itself, but how do I see this from a multiplication perspective?
\[L = \int\limits_{}^{}\sqrt(1+(\frac{ dy }{ dx})^2dx\]

Answer 3

Hmm I'm not sure how to think of arc length from a multiplicative perspective...
Honestly, I'm usually thinking of integration in terms of sums:
Adding up a whole bunch of pieces or rectangles or shapes or something.

So in the case of arc length, you're adding up a bunch of pieces of arc.
\(\large\rm \ell^2=dx^2+dy^2\)
(I can explain this relationship further if it doesn't make sense)
And solving for this piece of arc l,
\(\large\rm \ell=\sqrt{1+\left(\frac{dy}{dx}\right)^2}~~dx\)

So to me at least,

\(\large\rm \text{L = the sum of }\ell's\)

Answer 4

I don't think that really answers your question though :3 sorry heh

Answer 5

can you help me ^

Answer 6

@zepdrix Yes, thanks! I look at it that way too. It does make sense, but you can see that we actually force-simplified the original equation of\[\sqrt(dx^2 + dy^2) \rightarrow \sqrt{dx^2(1+\frac{ dy^2 }{ dx^2 })}\]
This finally gave us the form of something that fits all conditions for integration (you always need it multiplied by dx, right?)
\[\int\limits_{}^{}\sqrt{1+(\frac{ dy }{ dx })^2} dx\]
That's how we derived it in calc class, but it feels so forced, I don't know how to interpret this as multiplication!

Answer 7

say we want to find the length of this curve y=f(x) on (a,b)