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47 Class Note for MATH 230 at PSU

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This 2 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Pennsylvania State University taught by a professor in Fall. Since its upload, it has received 19 views.

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Date Created: 02/06/15
Sample Test 4 1 True or False Clearly indicate your Choice a T F 02lsfxy dx dy1502fxy dx dy b T F Let a the area of the region R in the zy plane lf ay k for all z and y in R7 then ay dA ka R 7r2 1 C T F Vol ofsphere x2y2221is given byV8 17r2rdrd6 0 0 2 27r 35in9 2 35in9 dT F p2s1n dpd6d 27rOO pzsina dpda 2 5 2 dsdy 0 3 a2 b 5 C 4 d e 10 f Does not Exist 3 Aljxlizzdydz a b i C i d 1 e 2 f 2X N H 4 The following integral de nes the volume of the region inside the sphere 2 y2 22 a2 above the xy plane over the region x2 y2 S 1 1 11712 dazixziyz dy dx 1 7W When this is converted to polar coordinates7 the resulting double integral has the form 27r 1 27r 1 27r 1 a r2 dr d0 b r a2 7 r2 d0 dr 0 7 dr d0 0 0 0 0 0 0 d 027r fol W d7 d9 e 027 017 Va 7 r2 dr d6 2 m2 5 The iterated integral ay dy dx is equivalent to 0 0 a 040 fzy dx dy b A4fzy dx dy C AZjg sy dx dy d O4fltwgtdx dy e 02fltwgtdx dy 10 11 12 13 14 15 16 17 18 6 6 Evaluate the following iterated integral Exact answers only No numerical approximations ey2 dy dx 0 m Express the following as a single iterated integral Do not evaluate it 2 m 4 47m dy dp dy dx 0 0 2 0 Find the COM of the triangle whose vertices are 007 017 20 and whose density is given by Wt 11 ky Set up but do not evaluate a double integral that gives the area of the surface on the graph of ay zey over the region R consisting of all z and y such that 0 S x S 4 and 0 S y S 10 Set up but do not evaluate a triple integral for the mass of the solid region Q bounded by the graphs of the equations below Assume that density is proportional to the distance from the origin Q 227y7 207 1107 3 z0 Suppose Q is the region in space above the xy plane inside the sphere 2 y2 22 16 Consider the integral 952 2 dV Qlt y a Set up but do not evaluate an iterated triple integral in cylindrical coordinates that could be used to evaluate this integral b Set up but do not evaluate an iterated triple integral in any co ordinates that nds the COM of Q Find the volume of the solid below 2 x2 y2 over the region R x2 y2 S 1 Use a double integral in polar coordinates to nd the volume ofthe solid of x7 y e wH over the region 2 y2 S 1 Show your work Exact answers only NO NUMERICAL APPROXIMATIONS Find area of parallelogram with vertices 004163 and 22 by attempting a change of variables via z 4u 7 1 and y 7 1Hiht The sides are parallel to y z and z 4y Using a change of variable7 compute x y2 sinx 7 y dA where R is the square with vertices R W70 lt37quot 713977139 g Hint use u xyw x7y the square have sidesz7y 7T z7y 07zy7rxy27r Bonus 1 Page 1041 Section Project Find the volume of the Wrinkled Sphere and the Bumpy Sphere Bonus 2 Page 1041 Question 46 Bonus 3 Find the volume of the four dimensinal ellipsoid 2 y2 222 1 2w2 1

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