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 area under y =  x in the first quadrant on the interval [0, 1] is revolved around the line y = -1? – A. 11  / 6 – B.  – C. 13  / 6 – D. 17  / 6 – E. 

Techniques of Integration (9/13/13) 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 & 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.

Integration By Parts Whereas substitution technique 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 Anti-differentiate 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 Using the same technique as for the ln(x) function, compute the antiderivative of arctan. 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!!