Second Examination. Tuesday, October 20, 1998.

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Problem1. % latex2html id marker 210 \( (\alph{enumi}) \) For the region shown in Fig. 1, decide whether to integrate with respect to \( x \) or \( y \). Also, draw a typical approximating rectangle and label its height and width. Finally, calculate the exact area of the region. Show all steps in the integration.















  
Figure: \( x=f(y)=y^{3}-y \) and \( x=g(y)=1-y^{2} \)
\begin{figure} \centering \includegraphics{182F98E2gr1.eps} \end{figure}

Problem2. % latex2html id marker 215 \( (\alph{enumi}) \) Find the volume of the solid obtained by rotating the hatched region in Fig. 2 around the \( x \)-axis. Sketch the solid and a typical approximating cylinder (disk). You may assume that the constant \( b \) is greater than 1. Show all steps in the integration.
  
Figure: \( y=f(x)=x^{2/3} \)
\begin{figure} \centering \includegraphics{182F98E2gr2.eps} \end{figure}

Problem3. % latex2html id marker 219 \( (\alph{enumi}) \) Find the exact value of the arclength of the curve having parametric equations \( x=t^{3} \) and \( y=t^{2} \), for \( 0\leq t\leq 1 \). Show all work.






















Problem4. % latex2html id marker 223 \( (\alph{enumi}) \) Suppose that \( {\displaystyle f(x)=\sin\left(\frac{\pi}{2}x\right)} \) and let \( m \) be the average value of \( f \) on the interval \( [0,2] \). Find the exact value of \( m \). Show all work.






















Problem5. % latex2html id marker 229 \( (\alph{enumi}) \) A spring scale in a warehouse has a 100 lb. calibration weight hanging from it. The scale accurately reads ``100'', and the pointer has moved a total of 6 inches from the zero reading. How much work was done against the spring when the weight was hung on the spring scale? Express your answer in units of ft-lb . Show all work. (You should assume that the spring satisfies Hooke's law).
Problem6. % latex2html id marker 230 \( (\alph{enumi}) \)
1.
Suppose that \( f(x)=\cos(x) \) for \( 0\leq x\leq\pi/2 \), and \( f(x)=0 \) for all other values of \( x \). (The graph of \( f \) is shown in Fig. 3 ) Explain why the function \( f \) is a probability density function.














2.
Suppose that \( X \) is a random variable having probability density function \( f \) as defined in the first part of this problem. Find the following probabilities: \( P[X<\pi/6] \) and \( P[X\geq \pi/6] \). Show all work.















  
Figure: The density \( y=f(x) \)

Problem7. % latex2html id marker 242 \( (\alph{enumi}) \) For what two values of the parameter \( k \) does the function \( y=e^{kx} \) satisfy the differential equation \( y''-2y'-3y=0 \)? Show all work.