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 (1642-1727)Sir Isaac Newton –Gottfried Wilhelm Leibnitz (1646-1716)Gottfried Wilhelm Leibnitz (1646-1716) 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)