Clicker Question 1 What is the volume of the solid formed when the curve y = 1 / x on the interval [1, 5] is revolved around the x-axis? – A.  ln(5) –

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11 The student will learn about: §4.3 Integration by Substitution. integration by substitution. differentials, and.

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Clicker Question 1 What is the volume of the solid formed when the curve y = 1 / x on the interval [1, 5] is revolved around the x-axis? – A.  ln(5) – B. 4  / 5 – C. 4 / 5 – D. 124  / 375 – E.  / 5

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. 17  / 6 – B.  – C. 5  – D. 13  / 6 – E. 

Techniques of Integration (9/14/12) 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, Constant Multiplier, and Reverse Power. There are no Product, Quotient, or Chain Rules. We need “techniques” rather than just rules. The first technique is 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 e x 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 Integrate 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 e x dx ?? Note x gets simpler when you take its derivative and e x ’s anti-derivative is no worse, so we try letting u = x and v = e x Then u = 1 and v = e x, so rebuild, using the Parts Formula:  x e x dx = x e x -  e x dx = x e x – e x + 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

A Few More... Remember, you have various techniques to try now.  x cos(x 2 ) dx ??  (x+4)  x dx ??  ln(x) dx ?? (Yes, we can get this one now! Hint: let v = 1)

Assignment for Monday For Monday, read Section 7.1 In that section do Exercises 3, 5, 9, 11 15, 23, 27, 64(b) and 65 (on 64(b), see the work already done on #15). Again, plan your weekend work time carefully. Don’t get behind in this course!!