# 2ab - MA 241 Test 2 Form B Spring 2007 Begin each new...

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Unformatted text preview: MA 241 Test 2 Form B Spring 2007 Begin each new problem on the top of a new page, back or front. NO calculators, No PDA, No cell phones , No notes, no etc! !! 1. (l0 points each) For each of the following, show every step required to set up the deﬁnite integrals that will solve the problems but do not integrate. a. Find the average value of the function on the interval x=l to x=10 for 3 ﬁx) =W- b. A uniform cable hanging over the edge of a tall building is 40 it long and weighs 50 lb. How much work is required to pull the cable to the top? c. A vertical dam has a semicircular gate as shown in the ﬁgure below, The density of water is 9800 Newtons per cubic meter. Find the hydrostatic force against the gate. i}2m water level d. Find the volume of the solid generated by revolving abOut the y-axis the region . . 3 bounded by the x-axis and y " 3x + x rmm x— 0 to x—S. e. For the lamina of density 9 formed by the region bounded by 3’ =V; and the x-axis ﬁ'om x=0 to x=8, ﬁnd the y-coordinate of the centroid. 2. (l4 points) Find the orthogonal trajectories for the family of curves 2 , 3 y a I“ (Note: this is not a set up, but SOLVE) Continue on next page 3. (14 points each) Solve the diﬁerential equations 2 ﬂ = , y(2) = 0 implicit solution a_ dx 2y + sm y ﬂ = —3y, y(0) = 8, explicit solution b. dx 4. ( 8 points) A population is modeled by the differential equation 92 = pg. 1.).) dt 100 a. What are the equilibrium solutions? b. For what values of P is the population decreasing? I did not give nor receive aid on this test. MA 241 Spring 2007 Test 2 solution key limits version ‘7; NW Problem 1 3. Find the arc length of the curve y = 933/2 from cc = 0 to a: = 5. Solution. First we compute d9 _ 3 1/2 din-2m And now we substitute into our formula 5 3 2 / 1+ (-xl/Z) d2: 0 2 Problem 1 b.A spring has a natural length of 1 meter. If a 24 N force is required to keep it stretched 3 meters beyond its natural length, how much work is required to stretch itfrom3mto4m? Solution. We ﬁrst must ﬁnd the spring constant via the formula F = km, where F is the force, It is the spring constant, and a: is the distance beyond the natural length. We have stretched it 3 meters beyond natural with 24 N, therefore, 24 = k a: 3 or, solving for k, k = 8. We now wish to ﬁnd the amount of work to stretch it from 3m to 4m. All our calculations are done With respect to the natural length, so we stretch it from 3 — 1 = 2 meters beyond natural to 4 — 1 = 3 meters beyond natural, so the integral is 3 W =/ 81min: 2 Problem 1 c. Find the volume of the solid generated by revolving about the y-axis the region bounded by the x—axis and y = 33: + 3:3 from cc = 0 to a: = 1. Solution. Since we are revolving this around the y—axis and we have a function giving y in terms of :13, it would be easiest to make our elements of area parallel to the axis of revolution. The formula is b / 27r * radius * height a: width Our radius is the distance from the axis of revolution, or just as, our height is simply our function 39: + 1:3 and the width is the tiny dz. Our limits of integration are given, so we have 1 / 27ra:(3x + x3)dx o Problem 1 d. The tank shown on the test is full 03 water. Given that water weighs 62.5 pounds per cubic foot, ﬁnd the work required to pump the water out of the tank. Solution. For the sake of simplicity, I will call the top-center of the tank the origin. That gives us an easier formula for for the size of the tank, since the walls can now be described via x2 + 3/2 = 25 for a 2 dimensional slice of the tank. Our formula topofwater / Density * (Distance to travel) * (Tiny volume of water) battanwfwater Looking from above, we see that our tank is circular, so our tiny volume is going to be 7rr2dy. The radius is simply the a: value for the wall of the tank, so 7‘2 is :62, and solving from above, we get 2:2 = 25 — y2. So, our tiny volume of water is «(25 — y2)dy. Since we are calling the top of the tank 0, then the bottom of the tank is ——5. We see that if we are y, which lies between —5 and 0, then we have —y feet to travel to get to the top. So, our distance is —y. The water ﬁlls the entire tank, from y = —5 to y = 0, so we now sum up /0 62.5(—y) * 1r(25 — y2)dy -5 Problem 1 e. For the lamina of density ,0 formed by the region bounded by y = 231/3 and the x—axis from a: = 0 to x = 0, ﬁnd the x-coordinate of the centroid. Solution. The bottom function here is the x-axis, that is, y = 0. We are given our limits of integration, so we simply substitute our values into the formula f: 551/3 * axis: 5;- = foe z1/3da: ...
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## This note was uploaded on 03/20/2008 for the course MA 141 and 24 taught by Professor Dempster/mccolum during the Spring '08 term at N.C. State.

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2ab - MA 241 Test 2 Form B Spring 2007 Begin each new...

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