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## Calculus 1

### Course: Calculus 1>Unit 6

Lesson 1: Accumulations of change introduction

# Introduction to integral calculus

The basic idea of Integral calculus is finding the area under a curve. To find it exactly, we can divide the area into infinite rectangles of infinitely small width and sum their areas—calculus is great for working with infinite things! This idea is actually quite rich, and it's also tightly related to Differential calculus, as you will see in the upcoming videos.

## Want to join the conversation?

• Something I don't really understand but have been "pretending" to understand in class is what d really means. I understand where to put d/dx and dy/dx, but what does dx or dy really mean? I have been told that it is "an infinitely small change in" but then what does "in relation to" mean in the definition of dy/dx??
• Leibniz introduced the d/dx notation into calculus in 1684. The "d" comes from the first letter of the Latin word "differentia", and it represents an infinitely small change, as you said, or "infinitesimal". The Greek letter delta is also used to represent change, as in Δv/Δt, so dv/dt is not a big stretch.

The "in relation to" or "with respect to" that you refer to is the quantity in the denominator, and is normally the independent variable. If you take the derivative of a function with respect to x, that would be for a function of x, and is written as d/dx. For a function of time, as I wrote above, dv/dt would be the derivative of the velocity with respect to time, meaning that the function is written as a function of time. The velocity (the dependent variable) changes with respect to time (the independent variable), and it's derivative is acceleration.

Hope that helps.
https://en.wikipedia.org/wiki/Leibniz%27s_notation
• Okay, so integration is basically finding the area under a curve and it is kind of like the opposite of differentiation and hence is called the derivative. So does it mean that when you take the derivative, you are actually breaking up the curve into rectangular components?
• When Sal used the new notation at , I got confused. What does the notation "dx" mean in ∫ f(x) dx? Does it mean "with respect to x"? Or the derivative of something?

edit: Does dx in this case represent an infinitesimally small delta x?
• The "dx" indicates that we are integrating the function with respect to the "x" variable. In a function with multiple variables (such as x,y, and z), we can only integrate with respect to one variable and having "dx" or "dy" would show that we are integrating with respect to the "x" and "y" variables respectively.
• so when doing any sort of integral problems, am I essentially finding the area under the curve between the upper limit and the lower limit? or what else do we use integral for?

I feel really behind in class because I can't keep track of what to do when they give me a problem. sometimes I have to take antiderivative then plug the upper limit and the lower limit then subtract; and sometimes I just plug in the upper and lower limit then subtract.
• Yes, finding a definite integral can be thought of as finding the area under a curve (where area above the x-axis counts as positive, and area below the x-axis counts as negative).

Yes, a definite integral can be calculated by finding an anti-derivative, then plugging in the upper and lower limits and subtracting.
• which class is appropriate to start learning calculus?
• Generally calculus (both differential and integral) is taught in junior and senior years (11th and 12th.) But there's absolutely no problem in learning it any time you want :)
I've learnt it during my freshman year (9th grade)!

I hope this helped!
• hey , i have a doubt. what do you actually mean by d/dx(sin x )=cos x and integral of sin x = -cos x ....i have been trying to understand but i couldnt ..can you please explain this
• sin x(d/dx)= cos x

cos x(d/dx) = -sin x

-sin x(d/dx) = -cos x

-cos x(d/dx) = sin x

You can just accept the fact sin x(d/dx) = cos x and
cos x(d/dx) = -sin x or take a look at the proof theorem for sin(x) d/dx = cos x which is on this site.

Note that if you carry out the calculations in degrees for the proof you will get sin x(d/dx)= pi/180 cos x not cos x. Hence pi radians was defined to be equal to 180 degrees which simplified the equation.

Note: Sometimes there might be theorems with proofs outside the scope of high school however nothing will stop you from applying the theorem. So if you wish take a look at the proof. However, it is okay if you don't understand it so long as you understand the theorem. You can always come back to the proof a couple years in the future if you are interested.
• When will we need to do antiderivatives vs definite integrals?
• Antiderivatives and integrals are the same exact thing.
• If dx becomes infinitely small, doesn't that mean that it moves closer and closer to zero and doesn't that mean that fx*dx will just approach zero?
• f(x) dx does approach zero, but the number of f(x) dx's approaches infinity. Adding together infinitely many infinitesimals generally gives you a finite value.
• so I think what my teacher has been trying to teach me for the last month just clicked but I want to make sure I got it right...the idea of integral calculus is taking the anti-derivative of various shapes under a curve whether it be rectangles or rectangles in order to determine the area under the curve as you approach infinity...please let me know if there are any flaws in the way that I am understanding it