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Clicker Question 1 What is the volume of the solid formed when area enclosed by the line y = x and the curve y = x2 is revolved around the x- axis? – – – – – A. / 6 B. 1 / 6 C. 2 / 15 D. / 15 E. 2 / 6 Clicker Question 2 What is the volume of the solid formed when the curve y = x on the interval [0, 1] is revolved around the line y = -1? – – – – – A. 11 / 6 B. 3 / 2 C. 5 D. 17 / 6 E. 13 / 6 Techniques of Integration (2/5/14) Should be called “techniques of anti-differentiation”. Finding derivatives involves “facts” and “rules”. It is a mechanical process. Finding anti-derivatives is not mechanical. The only rules are Sum & Difference and Constant Multiplier. There are no Product, Quotient, or Chain Rules. We need “techniques” rather than just rules. The first two techniques are algebraic manipulation and substitution (which tries to reverse the Chain Rule). Integration By Parts Whereas substitution techniques tries (if possible) to reverse the chain rule, “integration by parts” tries to reverse the product rule. Example: x ex dx ?? – – Substitution? No! Question: Can the integrand be split into a product of one part with a nice derivative and another part whose anti-derivative isn’t bad? Reversing the product rule If u and v are functions of x, then by the product rule: d/dx (u v) = u v + u v Rewrite: u v = d/dx (u v) - u v Antidifferentiate both sides, obtaining the Integration by Parts Formula: u v dx = u v - u v dx The hope, of course, is that u v is easier to integrate than u v was! Back to the Example x ex dx ?? Note x gets simpler when you take its derivative and ex’s anti-derivative is no worse, so we try letting u = x and v = ex Then u = 1 and v = ex , so rebuild, using the Parts Formula: x ex dx = x ex - ex dx = x ex – ex + C A quick check, which of course involves the product rule, shows this is right. Clicker Question 3 x cos(x) dx ? – A. ½ x 2 sin(x) + C – B. -½ x 2 sin(x) + C – C. x cos(x) – sin(x) + C – D. x sin(x) – cos(x) + C – E. x sin(x) + cos(x) + C Antiderivative of the natural log We now can figure out the antiderivative of the natural log function. Try letting u = ln(x) and v ' = 1. Assignment Hand-in #1 is due at class time on Friday. For Friday, read Section 7.1 and do Exercises 3, 5, 9, 11,15, 23, 27 and 64b and 65.