w05 - the depth of the water is 3 feet, what percentage of...

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Math 152 Workshop 5 Spring 2011 1. Calculate the area inside both of the ellipses x 2 3 + y 2 = 1 and x 2 + y 2 3 = 1. y 1.0 0.0 -1.0 0.0 x 1.5 1.5 0.5 -0.5 0.5 -1.5 1.0 -0.5 -1.0 -1.5 2. (a) Suppose A is a positive real number and m A is the average value of (sin( Ax )) 3 on the interval [0 , 2]. Compute m A . Note The answer will have several terms and will not be simple. (b) What is lim A →∞ m A ? Note This answer should be simple. Explain briefly why it is correct. You may refer to graphs of functions if that is helpful. 3. (a) Suppose that m and n are integers. Compute R 2 π 0 ( cos( mx ) )( cos( nx ) ) dx . (Be careful: there will be two different results, one when m = n and one when m 6 = n .) (b) Suppose f ( x ) = A cos( x ) + B cos(2 x ) + C cos(3 x ), and that you also know R 2 π 0 f ( x ) cos( x ) dx = 5 ; R 2 π 0 f ( x ) cos(2 x ) dx = 6 ; R 2 π 0 f ( x ) cos(3 x ) dx = 7 . Find A and B and C . Note The ideas of this computation are used often with Fourier series, a standard method of analyzing periodic phenomena. 4. An oil tank has the shape of a cylinder whose diameter is 4 feet. It is mounted so that the axis of the cylinder is horizontal (the circular cross-sections of the cylinder are vertical). If
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Unformatted text preview: the depth of the water is 3 feet, what percentage of the total capacity of the tank is lled? After drawing a picture and setting up this problem, solve it three ways: (a) Use elementary geometry (compare areas of circular sectors). (b) Express the answer in terms of a denite integral, then obtain an approximate numerical value for the integral using the fnInt( function on your calculator. (c) Evaluate the integral in b) exactly in terms of elementary functions using a trig substitu-tion, then obtain approximate numerial values for these functions using your calculator. 5. (a) Find R e 2 x e 2 x +1 dx. (b) Find R e x e 2 x +1 dx. Comment These antiderivatives may appear similar, but dierent methods are needed....
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