WebWork 3 - Jake Sieger Section SEC2 0868 Homework Set...

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Jake Sieger - Section SEC 2 0868 Homework Set Homework3 due 10/19/2011 at 11:55pm EDT This set covers sections 6.1-6.3 of the text. You may need to give 4 or 5 significant digits for some (floating point) numerical answers in order to have them accepted by the computer. 1. (1 pt) The area A of the region D bounded by graphs y = 3 x 2 + 3 , y = - 3 x , x = - 3 , and x = 1 can be computed as an integral 1 - 3 x y y = 3 x 2 + 3 y = - 3 x D R b a f ( x ) dx , where a = b = f ( x ) = The area A = 2. (1 pt) The area A of the region D bounded by graphs y = 1 - 2 x 2 , and y = - 2 x - 11 can be computed as an integral x y y = 1 - 2 x 2 y = - 2 x - 11 D R b a f ( x ) dx , where a = b = f ( x ) = The area A = 3. (1 pt) Note: You can get full credit for this prob- lem by just answering the last question correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get par- tial credit. The integral R 4 - 1 9 x 2 - x 3 - 18 x dx MUST be evalu- ated by breaking it up into a sum of three integrals: Z a - 1 9 x 2 - x 3 - 18 x dx + Z c a 9 x 2 - x 3 - 18 x dx + Z 4 c 9 x 2 - x 3 - 18 x dx where a = c = R a - 1 9 x 2 - x 3 - 18 x dx = R c a 9 x 2 - x 3 - 18 x dx = R 4 c 9 x 2 - x 3 - 18 x dx = Thus R 4 - 1 9 x 2 - x 3 - 18 x dx = 4. (1 pt) Find a positive real number b such that Z 2 π 0 | 8cos ( x )+ b sin ( x ) | dx = 36 To solve this problem, we first rewrite the expression 8cos ( x )+ b sin ( x ) in the form A sin ( x + α ) where A = (your answer should be a function of b ) and α is some angle between 0 and 2 π . Using the trig identity sin ( θ + 2 π ) = sin ( θ ) we then find that R 2 π 0 | A sin ( x + α ) | dx = (again your answer should be a function of b ). Using the given value of the integral and solving we find that b = 1
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5. (1 pt) Find the area between the curves: y = x 3 - 15 x 2 + 50 x and y = - x 3 + 15 x 2 - 50 x 6. (1 pt) Note: You can get full credit for this prob- lem by just answering the last question correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get par- tial credit. Find the area bounded by the two curves: x = 100000 ( 10 y - 1 ) x = 100000 10 y - 1 8 y The appropriate definite integral for computing this area has integrand ; lower limit of integration = ; and upper limit of integration = This definite integral has value = This is the area of the region enclosed by the two curves.
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