Intuitive ways to think of integration and second derivative

[Maddie1609]

Hi,

I feel sometimes when I'm doing calculus I lose the logic and intuition behind what I'm doing, especially when integrating. I have yet to find a way to think about it in a way it makes sense to me why the definite integral would tell us the area under a curve. Same with why the second derivative would show the concavity. Does anyone have a good explanation, video or book that could help me think about it more intuitively? I don't like just applying methods without knowing what I'm actually doing

 

[RUber]

One definition of the integral is the sum of very small rectangles having width (dx) and height f(x). Using the rectangle definition, the area is the most intuitive understanding of a height times the width.
$\int_a^b f(x) dx = \lim_{dx \to 0} \sum_{j=0}^N f(x_j) * dx, \, N = (b-a)/dx, \, x_j = a + j*dx$
Of course there are other definitions of integrals, but this is a pretty commonly used one.

For the derivative, is it intuitive enough to think of the first derivative as the slope?
If so, the second derivative is the slope of the slope.
If the first derivative is positive, the function is going up. If the second derivative is also positive, the rate at which the function is going up is also increasing. This gives a concave-up type function.
If the first derivative is positive and the second derivative is negative, the rate at which the function is going up is decreasing. This give a concave-down shape to the graph of the function.

I will look around for some visuals that make these points clearer.

 

[Mark44]

Hi,

I feel sometimes when I'm doing calculus I lose the logic and intuition behind what I'm doing, especially when integrating. I have yet to find a way to think about it in a way it makes sense to me why the definite integral would tell us the area under a curve.

Just about every calculus textbook leads up to the definite integral showing a picture similar to the following.

(See https://en.wikipedia.org/wiki/Integral)
The textbooks usually define the Riemann integral as the limit of a sum of area elements. Haven't you ever seen something like this?

Same with why the second derivative would show the concavity. Does anyone have a good explanation, video or book that could help me think about it more intuitively?

Do you have a feel for the meaning of the first derivative -- what it says about the graph of some function?

I don't like just applying methods without knowing what I'm actually doing

 

[Legolaz]

Hi,

I feel sometimes when I'm doing calculus I lose the logic and intuition behind what I'm doing, especially when integrating. I have yet to find a way to think about it in a way it makes sense to me why the definite integral would tell us the area under a curve. Same with why the second derivative would show the concavity. Does anyone have a good explanation, video or book that could help me think about it more intuitively? I don't like just applying methods without knowing what I'm actually doing

Hi there, Maddie.
We have the same sentiments way back my undergraduate years. I tend to solve or study things I can not realistically appreciate, thus I think of calculus as out of this world concept and certainly one of my greatest burden and hardships way back then.

Lately, I learned that is indeed a useful tool to grasp, especially dealing with sciences and engineering.

First, you need to have a mind set or convinced yourself, that calculus is not that hard - believe me this will help a lot.

Derivative is nothing but taking differences, or portions of any physical quantities (length, time, volume, mass etc.) in simple terms, on the other hand, integral or integration is just summing up all the portions on interest.

Derivative and integral comes together to provide us tools to measure lengths of a strip, surface areas, volumes (both with irregular and regular geometries).

The first derivative f'(X) of any equation or function f(x) is a function tangent to the original function . The second derivative f"(x) of function f(x) is the line(if 2d) or surface(if 3d) perpendicular to the first derivative function.
What is the importance of these? These represents the analogy of vectors (Force, Velocity, Magnetic field, etc) in physics or natural world that we could analyze them and design for use involving these parameters.

 

[Ssnow]

I found this intuitive video (on you tube) about the definition of definite integral:



in any case I suggest a calculus book for a rigorous explanation.