First Name:  Last Name:

Evaluate the following limits:

1. Find the surface area of the surface of rotation of the curve \(y = x\) between \(x = 0\) and \(x = 1\).

a.  \(\pi \)

b.  \(\sqrt{2}\pi\)

c.  \(2 \pi\)

d.  \(2\sqrt{2} \pi)\)

e.  \(4 \pi\)

10. Find the total work done if you push a rock from \(x = 0\) m to \(x = 4\) m if you're pushing with force \(F(x) = x\) N.

a.  \(2 J \)

b.  \(4 J\)

c.  \(8 J\)

d.  \(16 J\)

e.  \(32 J\)

For the next two problems, we consider a 2.00 m long metal bar which has density \(\rho(x) = 12.0 x^2 + 18.0\) kg/m between \(x = 0\), and \(x = 2\).

11. Find the total mass \( m\) of the metal bar.

a.  48 kg

b.  58 kg

c.  68 kg

d.  78 kg

e.  85 kg

12. Find the center of mass \(\overline{x}\) of the metal bar.

a.  1.00 m

b.  1.17 m

c.  1.20 m

d.  1.24 m

e.  1.35 m

13. \(\int \frac{2}{(x-1)^4} dx \)

a.  \(2\ln(x-1)^4 + c\)

b.  \(8\ln\mid x-1 \mid\)

c.  \(-\frac{2}{3(x-1)^3}+c\)

d.  \(-\frac{8}{(x-1)^5}+c\)

e.  \(8\ln(x-1)^4 + c\)

14. \(\int x^4 \ln x dx \)

a.  \(\frac{1}{5}x^5\ln x -\frac{1}{25}x^5+c\)

b.  \(\frac{1}{6}x^6\ln x - \frac{1}{6}x^6+c\)

c.  \(\frac{1}{6}x^6\ln x+c\)

d.  \(x^3+4x^3\ln x + c\)

e.  \(\frac{1}{6}x^6\ln x - \frac{1}{36}x^6+c\)