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Clicker Question 1 What is  cos 3 (x) dx ? – A. ¼ cos 4 (x) + C – B. -3cos 2 (x) sin(x) + C – C. x – (1/3) sin 3 (x) + C – D. sin(x) – (1/3) sin 3 (x)

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Presentation on theme: "Clicker Question 1 What is  cos 3 (x) dx ? – A. ¼ cos 4 (x) + C – B. -3cos 2 (x) sin(x) + C – C. x – (1/3) sin 3 (x) + C – D. sin(x) – (1/3) sin 3 (x)"— Presentation transcript:

1 Clicker Question 1 What is  cos 3 (x) dx ? – A. ¼ cos 4 (x) + C – B. -3cos 2 (x) sin(x) + C – C. x – (1/3) sin 3 (x) + C – D. sin(x) – (1/3) sin 3 (x) + C – E. sin 2 (x) cos(x) + C

2 Clicker Question 2 What is volume generated when the area under y = tan(x) between x = 0 and x =  /4 is revolved around the x-axis? – A. 1 –  /4 – B. 1 +  /4 – C.  –  2 /4 – D.  +  2 /4 – E.  –  2 /8

3 Trig Substitutions (2/10/14) Motivation: What is the area of a circle of radius r ? Put such a circle of the coordinate system centered at the origin and write down an integral which would get the answer. Can you see how to evaluate this integral?

4 Replacing Algebraic With Trigonometric Recall that d/dx(arcsin(x)) = 1/  (1 – x 2 ) Hence  1 /  (1 – x 2 ) dx = arcsin(x) + C This leads to the idea that integrands which contain expressions of the form (a 2 – x 2 ) (where a is just a constant) may be related to the sin function. Thus in the expression above we make the substitution x = a sin(  ) (so dx = a cos(  )d  ).

5 An example What is  1 /(1 – x 2 ) 3/2 dx ? Regular substitution? Well, try a “trig sub”. Let x = sin(  ), so that dx = cos(  ) d . Now rebuild the integrand in terms of trig functions of . Can we integrate what we now have?? (Yes, think back!) Finally, we must return to x for the final answer.

6 Back to the circle Try a trig substitution. Since this is a definite integral, we can eliminate x once and for all and stick with  (so we must replace the endpoints also!). Look familiar? We’ve now proved the most famous formula in geometry.

7 Other trig substitutions Recall that d/dx(arctan(x)) = 1/ (1 + x 2 ) Hence  1 / (1 + x 2 ) dx = arctan(x) + C This leads to the idea that integrands which contain expressions of the form (a 2 + x 2 ) (where a is just a constant) may be related to the tan function. Thus in the expression above we make the substitution x = a tan(  ) (so dx = a sec 2 (  )d  ).

8 Assignment for Wednesday Make sure you understand and appreciate the derivation of the area of a circle. Assignment: Read Section 7.3 and do Exercises 1, 2, 4, and 9.


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