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Unlocking Calculus. Getting started on Integrals. The Integral. What is it? The area underneath a curve Why is it useful? It helps in Physics, Chemistry, Engineering Total energy of a system, total force, mass … It is one step closer to Truth!. Sir Isaac Newton (1642-1727).

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unlocking calculus

Unlocking Calculus

Getting started on Integrals

the integral
The Integral
  • What is it?
    • The area underneath a curve
  • Why is it useful?
    • It helps in Physics, Chemistry, Engineering
      • Total energy of a system, total force, mass …
    • It is one step closer to Truth!
history who invented the integral
Sir Isaac Newton (1642-1727)

Gottfried Wilhelm Leibnitz (1646-1716)

Newton v. Leibnitz

HistoryWho “invented” the integral?

Calculus

notation
Here’s what we write:

Here’s what we mean:

The area from a to b underneath the function f(x) with respect to x.

Huh?

Notation
for starters
What does it mean?

The area from 2 to 4 underneath the function f(x)= x with respect to x

So the area of the square is 4 and the triangle is 2. Thus the answer is 6.

For Starters
the next step
Here’s the graph

We want to find the areaunder the graph from 1 to 3

But since this is a curve,we don’t have an exact formula for the area…

So we should just give up!

Not yet…

The Next Step
just get close
We can approximate by using rectangles.

The more rectangles we use,the closer the approximation.

In, fact if we used an infinitenumber of rectangles, evenlyspaced, it would be exact!

But I can’t draw that manyrectangles

And you can’t add that many either!

Just get close!
calculus to the rescue
Calculus to the rescue
  • Calculus has found certain formulae that help solve such problems
  • We’ll unlock two of these formulae right now.
    • The Power Rule
    • The Addition Rule
the power function
The Power Function
  • A power function is any function with x raised to some exponential power.

These are just a few examples

the power rule
Here’s the general form:

_____________________

In words, the integral of (x to the n) from a to b with respect to x is (x to the n+1) over (n+1) evaluated from a to b, which is [(b to the n+1) over (n+1)] minus [(a to the n+1) over (n+1)].

The Power Rule
example
Example
  • Here’s one for starters
addition of functions
Few functions are simple power functions. Many involve addition of two or more simple power functions

So, there’s a rule to handle such common functions.

Addition of Functions
addition rule
The general rule is written

_______________ ________ ________

(in words) The integral from a to b of the function [g(x)+h(x)] with respect to xisthe integral from a to b of g(x) with respect to xplusthe integral from a to b of h(x) with respect to x.

OR the integral of a sum is the sum of the integrals (and vice versa)

Addition Rule
example14
Example
  • Don’t forget the previous lessons, they’re important now!
the end
There are many more kinds of functions, and thus many more integrating rules, but this is the intro.

Click here to view a summary of related websites

Remember, mastering the basics is the key to unlocking Calculus (and everything else)

The “End”