Test 3

1. Find the length of the curve

$y=\frac{1}{6}{(x^2+4)}^{3/2 }, 0 \le x \le 2$

2. Find the area of the surface obtained by rotating the curve about the x-axis

$x=\frac{1}{3}{(y^2+2)}^{3/2}, 1 \le y \le 3$

3   Find the length of the curve

$x=\frac{y^4}{8}+\frac{1}{4y^2}$

  ,

4. Find the length of the curve for the interval 

$y = {\int }_1^x\sqrt{t^3–1dt}$

5. Find the area of the surface obtained by rotating the curve about the y-axis.

$x=y^3, 0 \le y \le 1$

6.  Find the area of the surface obtained by rotating the curve about the x-axis.

$y=x^3 , 0 \le x \le 3$

7.   Find a polar equation for the curve represented by the given Cartesian equation

$x^2=9y$

8.  Find the polar equation for the curve represented by the given Cartesian equation

$x + y = 4$

9.  Find the length of the curve

$x=3t^2 , y= 2t^3, 0 \le t \le 1$

10 Find the length of the polar curve

$r = 7 \cos \theta , 0 \le \theta \le 3\mathrm{\pi }/4$

11. Find an equation of the tangent to the curve at the point corresponding to the value of the parameter

$x = t\sin t, y = t\cos t; t = 11\mathrm{\pi }$

12. . Find an equation of the tangent line to the given curve at the point corresponding to the specified value of the parameter.

13.  Find

14. Sketch the polar curve and show the axis of symmetry

15 . In the circuit shown in the figure, a battery supplies a constant voltage of 40 V, the inductance is 2 H, the resistance is 10 Ω, and I ( 0 ) = 0. Find the
current after 1 s