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# Calculus and Vector Analysis MATH 230

Penn State

GPA 3.88

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This 0 page Class Notes was uploaded by Elaina Osinski on Sunday November 1, 2015. The Class Notes belongs to MATH 230 at Pennsylvania State University taught by Staff in Fall. Since its upload, it has received 11 views. For similar materials see /class/233002/math-230-pennsylvania-state-university in Mathematics (M) at Pennsylvania State University.

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Date Created: 11/01/15

Sample Test 4 1 True or False Clearly indicate your Choice 2 5 5 2 a T F 0 1 fltzygtdx dy1 0 fltzygtdx 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 02 dx dy a 2 b 5 C 4 d e 10 f Does not Exist 3 Aljxlizzdydz a i b i C i d 1 e 2 f 2X 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 03 5 00 p H 0 H H 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 z3 z0 7 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 2y2 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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