Math150Sols5S11

# Math150Sols5S11 - QUIZ ON CHAPTER 5 SOLUTIONS MATH 150 –...

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Unformatted text preview: QUIZ ON CHAPTER 5 SOLUTIONS MATH 150 – SPRING 2011 – KUNIYUKI 105 POINTS TOTAL, BUT 100 POINTS = 100% Show all work, simplify as appropriate, and use “good form and procedure” (as in class). Box in your final answers! No notes or books allowed. A scientific calculator is allowed. Your final answers must not consist of any negative exponents or compound fractions. 1) Evaluate the following integrals. Simplify as appropriate. (46 points total) a) 5 w 2 ¡ 7 w + w w dw ¢ (9 points) = 5 w 2 w ¡ 7 w w + w w ¢ £ ¤ ¥ ¦ § dw ¨ = 5 w ¡ 7 + 1 w ¢ £ ¤ ¥ ¦ § dw ¨ = 5 w ¡ 7 + w ¡ 1/ 2 ( ) dw ¨ = 5 w 2 2 ¢ £ ¤ ¥ ¦ § ¡ 7 w + w 1/ 2 1 / 2 + C = 5 2 w 2 ¡ 7 w + 2 w + C , or 5 w 2 ¡ 14 w + 4 w 2 + C b) csc x sin x dx ¡ (5 points) = csc x ( ) 1 sin x ¡ ¢ £ ¤ ¥ ¦ dx § by a Reciprocal Identity ( ) = csc x ( ) csc x ( ) dx § = csc 2 x dx § = ¨ cot x + C , or C ¨ cot x c) 3sec 2 x tan 7 x dx ¡ (7 points) Let u = tan x ¡ du = sec 2 x dx 3sec 2 x tan 7 x dx ¡ = 3 sec 2 x tan 7 x dx ¡ = 3 u 7 du ¡ = 3 u 8 8 ¢ £ ¤ ¥ ¦ § + C = 3 8 tan 8 x + C d) 2 + x ( ) 6 x dx ¡ (8 points) Let u = 2 + x , or 2 + x 1/ 2 ¡ du = 1 2 x ¢ 1/ 2 dx du = 1 2 x dx ¡ Can use: 1 x dx = 2 du £ ¤ ¥ ¦ § ¨ 2 + x ( ) 6 x dx ¡ = 2 + x ( ) 6 ¢ 1 x dx , or ¡ 2 2 + x ( ) 6 2 x dx ¡ by Compensation ( ) , or 2 2 + x ( ) 6 ¢ 1 2 x dx ¡ = 2 u 6 du ¡ = 2 u 7 7 £ ¤ ¥ ¦ § ¨ + C = 2 7 2 + x ( ) 7 + C e) 9 ¡ x 2 3 ¢ dx (5 points) (Hint: Do not use the Fundamental Theorem of Calculus.) Geometric argument Graph f , where f x ( ) = 9 ¡ x 2 and ¡ x ¡ 3. The graph of f is a quarter of the circle of radius 3 centered at 0, 0 ( ) : y = 9 ¡ x 2 y 2 = 9 ¡ x 2 y ¢ ( ) x 2 + y 2 = 9 y ¢ ( ) The function f is nonnegative on 0, 3 [ ] , so the value of the given definite integral is equal to the area under the graph of f (and above the...
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Math150Sols5S11 - QUIZ ON CHAPTER 5 SOLUTIONS MATH 150 –...

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