Quiz #4 Math 181 Name: 10/15/20 Show all your work clearly. No Work, No Credit. 1. The region bounded by 23 ; 0y x x y � is rotated about the line 4x . Sketch and then set up the integral(s) for its volume. (5 pts) 2. Evaluate the following improper integrals: (Indicate if it’s convergence / divergence after the evaluation)

a) � �� �2/3/5

1ln 5e

dxx x

f

³ (4 pts)

b) 23

5 32 8x dxx �

³ (4 pts)

c) � �13

2

tan 51 25

xdx

x

�f

�f �³ (4 pts)

3. a) A 300 – m – cable that weight 600N is attached to a 1500 – N – elevator. Determine the total work needed to pull both cable and the elevator 300m up. (5 pts) b) It’s required a work of 50 ft - lb to stretch a spring 10in beyond its natural length. How much work is done to stretch the spring another 5in? (5 pts)

4. Test for convergence / divergence (Indicate clearly what theorem you are using)

a) � �

40

1sin 2

dxx x

S

³ (4 pts)

b) 3 5 3

3 22

3 2 13 2 4x x dxx x

f � �� �³ (4 pts)

5. Find the arc – length of the curve: � �1 3 1 93

x y y for y � d d (5 pts)

6. Determine the area of surface generated by revolving the curve 3/2 1/213

x y y � about the y – axis for

1 3yd d (5 pts)

7. Calculate the total hydrostatic a circular plate of radius 3m is submerged vertically under water 100m from its bottom. Calculate the total hydrostatic force on the plate. (5 pts)