math 2008 - Name 1(20 points Estimate the double integral R...

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Name: 1 (20 points). Estimate the double integral RR R cos( y/x ) dA , where R is the rectangle [0 , 2] × [0 ,π/ 2] using a double Riemann sum with four sub-rectangles. Use the upper right corner of each sub- rectangle for the sample point. It’s probably best to draw the rectangle R and the four sub-rectangles. The sub-rectangles are [0 , 1] × [0 ,π/ 4] , [0 , 1] × [ π/ 4 ,π/ 2] , [1 , 2] × [0 ,π/ 4] and [1 , 2] × [ π/ 4 ,π/ 2]. Each of them has area π/ 4 (one fourth the area of R ). The upper right corners are (1 ,π/ 4) , (1 ,π/ 2) , (2 ,π/ 4) and (2 ,π/ 2). The corresponding function values are cos( π/ 4) , cos( π/ 2) , cos( π/ 8) and cos( π/ 4). The double Riemann sum is π 4 ( cos( π/ 4) + cos( π/ 2) + cos( π/ 8) + cos( π/ 4) ) . 2 (20 points). Find the volume of the region bounded by the sphere x 2 + y 2 + z 2 = 1 and the cone z = p 3( x 2 + y 2 ). This region is more simply described in spherical coordinates: the equation x 2 + y 2 + z 2 = 1 is just ρ = 1 and
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This note was uploaded on 01/26/2012 for the course MATH 2008 taught by Professor Lucy during the Fall '11 term at Carleton CA.

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math 2008 - Name 1(20 points Estimate the double integral R...

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