Quiz #4 Math 181 Name: 4/15/21 Show all your work clearly. No Work, No Credit. 1. The area bounded by 2 0 0xy e and y for x−= = ≥ is rotated about the x- axis. Calculate its volume. (5pts)

2. Find the arc – length of the curve: 4/3 2/33 3 5 1 84 8

y x x for x= − + ≤ ≤ (5 pts)

3. Evaluate the following improper integrals: (10 pts)

a) ( )

3

42 2 9

x dxx −

∫

b) ( ) ( )2 10

11 4 tan 2 1

dxx x

∞

−+ +∫

4. Solve the differential equations: 3 2

3 2 1

x y

x y

dy edx e

− +

− += ( )0 2y = (5 pts)

5. Find the center of mass of the region bounded by 2 4 2y x x and y x= − = (5 pts)

6. A tank contains 20 gallons of water which is dissolved 5lbs of salt. A solution containing ½ lb of salt flows into the tank at a rate of 4 gal/min, and the well – stirred mixture flows out at the same rate of 4 gal/min. Set up an IVP (Initial Value Problem) for the amount of salt in the tank and then determine the amount of salt in the tank after 30 minutes. (10 pts)

7. The following tank has a rectangular window of dimension 2ft by 4ft as shown and it can withstand a hydrostatic force of 1,500 lbs. To what level can the tank be filled with water without exceeding the windows design limitation? (10 pts)

7. Sketch the graph with direction of the following parametric equation: ( )( )

3cos 2 1;

2sin 2 3

x tt R

y t

= −∈

= + (5 pts)