Unlocking Calculus Getting started on Integrals

Unlocking Calculus 1 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!

Unlocking Calculus 2 History Who “invented” the integral? –Sir Isaac Newton ( )Sir Isaac Newton –Gottfried Wilhelm Leibnitz ( )Gottfried Wilhelm Leibnitz ( ) Newton v. Leibnitz Calculus

Unlocking Calculus 3 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?

Unlocking Calculus 4 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.

Unlocking Calculus 5 The Next Step  Here’s the graph  We want to find the area under 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…

Unlocking Calculus 6 Just get close!  We can approximate by using rectangles. –The more rectangles we use, the closer the approximation. –In, fact if we used an infinite number of rectangles, evenly spaced, it would be exact! –But I can’t draw that many rectangles –And you can’t add that many either!

Unlocking Calculus 7 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

Unlocking Calculus 8 The Power Function  A power function is any function with x raised to some exponential power. These are just a few examples

Unlocking Calculus 9 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)].

Unlocking Calculus 10 Example  Here’s one for starters

Unlocking Calculus 11 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.

Unlocking Calculus 12 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 x is the integral from a to b of g(x) with respect to x plus the 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)

Unlocking Calculus 13 Example  Don’t forget the previous lessons, they’re important now!

Unlocking Calculus 14 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 websiteshere  Remember, mastering the basics is the key to unlocking Calculus (and everything else)

Unlocking Calculus 15 The Plan