Math 1B Calculus Test 3 Spring 10 Name Write all responses on separate paper. Show your work for credit.

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1 ath B Calculus Test Spring Name Write all responses on separate paper. Show our work for credit.. Determine whether each integral is convergent or divergent. Evaluate those that are convergent. a. / 4 d b. ln d. Use the comparison theorem to determine whether the integral is convergent or divergent: e d. Evaluate the integral ( ) sin n e d in terms of n. = + 6, Find the length of the curve ( ) / 5. Find the length of the curve = cos t dt, π. 6. Find the surface area generated b rotating about the ais the curve = sin() for π. 7. Find the surface area of surface generated b rotating the curve = - about the -ais for. 8. A circular gate in an irrigation canal has diameter 5 meters at the top. If the canal is filled to a depth of meters below the top of the gate, find the hdrostatic force on one side of the gate. 9. Find the centroid of the region bounded b = sin(π) and = ( ), ½.. Find the moments and center of mass of the sstem of objects that have masses 4, 7 and at the points (,), (,) and (, ), respectivel.. Lengths of human pregnancies are normall distributed with mean 68 das and standard deviation 5 das. What percentage of pregnancies last between 5 das and 8 das? Hint: use the ( μ) / ( σ ) normal probabilit densit function: f ( ) = e σ π. Let f ( ) k( ) = if and ( ) f = if < or >. a. For what value of k is f() a probabilit densit function? b. For that value of k, find P(X ) c. Find the mean.

2 ath B Calculus Test Solutions Spring. Determine whether each integral is convergent or divergent. Evaluate those that are convergent. a. / 4 d b. ln d π / secθ tanθ d = sec / d θ = d θ θ / 4 sec θ π / = ln secθ + tanθ = ln + So it s covergent and we know it s value. Compare < = < 4 + on [½,] to see the integral is convergent since d = = / / ln b u u u u b lim lim d ue du ue e du e = = + = = b b. Use the comparison theorem to determine whether the integral is convergent or divergent: e d e e < d < d < d, = du = lim u = + u b b therefore the integral is convergent. You can also do i.b.p here.. Evaluate the integral ( ) sin n e d in terms of n. sin ( ) = sin ( ) + cos ( ) cos( ) sin ( ) = n e d e n n n e d ne n n n e d n ( ) sin( ) sin( ) + n n e d= n n e d= + n = + 6, Find the length of the curve ( ) / = + + = + + = = 9 9 / ' ( 6) ( ' ) ( 6) L= + ( ' ) d= ( ) d + = + = 9

3 5. Find the length of the curve = cos tdt, π. ( ) π π π/ ' = cos + ' = cos L = cos d = cos d = cos d = 6. Find the surface area generated b rotating about the ais the curve = sin() for π. First substitute u = cos() so that du = cos()d and >u> then reverse the bounds and substitute u = tanθ so that du = sec (θ)dθ. Now use smmetr about the -ais and double the integral from θ π/4. π π/4 π sin + cos d = π + u du = π sec θ dθ π /4 ( ) π /4 ( ) ( ) = π secθtanθ + ln secθ + tanθ = π + ln + 7. Find the surface area of surface generated b rotating the curve = - about the -ais for πrds = π + d = π d 6 5. This is worth a trig substitution sec θ = tanθ ields π d = π d 5 θ but there doesn t seem to be a nice wa to substitute for the so there isn t a nice antiderivative, outside of the Elliptic functions. So tr 4 + integrating over : πrds = π + d = π d 4. Unfortunatel, this is also difficult because of the lack of an elementar antiderivative. In a situation like this, we are forced to emplo numerical approimations. 8. A circular gate in an irrigation canal has diameter 5 meters. If the canal is filled to a depth of meters below the top of the gate, find the hdrostatic force on one side of the gate. Center the coordinate sstem at the center of the gate. Then weight densit * depth * element of area = 98 * ( ) * sqrt(5 4 )

4 Thus the total fluid force against the gate is ( ) 5 4d= 98 ( ) 5 4d 5 4d arcsin. arcsin cos θ 5 4d= cos θdθ = dθ.5 π/ π/ Now, arcsin. 5 θ sinθcosθ 5 π 6 = + = + arcsin π /.5 4 / and 5 4 d = udu = ( 4) = So, all together we ve got.5 75 π 6 75π 98 arcsin arcsin = Find the centroid of the region bounded b = sin(π) and = ( ), ½. The mass is = sin ( π ) ( ) ( π ) / / cos d= + = + = + π π 4 8 π / / / ( π ) ( ) ( π ) ( ) = sin d= sin d d / / / / 4 ( π ) sin = cos( π) + cos( π) d π π = π = + π 9 / / / cos( π ) 4 = sin ( π ) ( ) d d d = + / / 5 4 ( π ) sin = + = + = 4π So π 9 + π = = =.7 and π = = =.7 + 9π + 6π + + π π π. Find the moments and center of mass of the sstem of objects that have masses 4, 7 and at the points (,), (,) and (, ), respectivel. = =. = m i i = ( 4) + ( 7) + ( ) =. i= m 4 ( ) 7 ( ) ( ) so = = + = i i i= 6 = = and = =. Lengths of human pregnancies are normall distributed with mean 68 das and standard deviation 5 das. What percentage of pregnancies last between 5 das and 8 das? Hint: use the ( μ) / ( σ ) normal probabilit densit function: f ( ) = e σ π

5 5 π 8 5. Let f ( ) k( ) e ( ) 68 /45 d = if and f ( ) = if < or >. a. For what value of k is f() a probabilit densit function? 4 k k d= k = = k = ( ) b. For that value of k, find P(X ) Clearl, this is. c. Find the mean ( ) d = 4 = 5 5