I’m studying and need help with a Calculus question to help me learn.

Hello,

Please solve with out a calculator.

Circle the final answer.

1. Find the arc length functions for y = arcsin (*x*) + *√*1 *− **x*2 with starting point (0, 1).

2. Find the exact length of the curve x =^{ }^{1}*√**y*(*y **− *3), 1 *≤ *y *≤ *9.

3. An oceanographer is testing the acidity of the Atlantic ocean by placing a circular measuring device on the vertical face of the continental shelf. The measuring device has a radius of 1 meter and is placed so the top of the measuring device is exactly 125 meters below the surface.

Calculate the hydrostatic force on the measuring device. Use 1024 kg/m^{3} for the density of the seawater in the Atlantic ocean, and 9.81m/s^{2} for the force of gravity.

4. (a) Find the centroid of the region bounded by y =^{ }__ ^{1}__ x +

^{ }

__, y = x__

^{1}^{3}, and x =

^{ }

__.__

^{1}2 2 4

(b) Find the centroid of the region bounded by (x – 2)^{2} + (y + 3)^{2} = 25.

5. Find the exact area of the surface obtained by rotating the curve y = x^{3}, 0 *≤ *x *≤ *2, about the x-axis

6. Eliminate the parameter to sketch the curve:

1

*x *= sin *θ,*

2

1

*y *= cos *θ,*

2

*−**π **≤ **θ **≤ **π*

7. Let x = *√*2 t^{2} + 6, y = __ ^{1}__ t

^{3}–

__t.__

^{1}2 3 2

- Find dy/dx.
- Find d
^{2}y/dx^{2}.

- Find the length of the curve for
*−*1*≤**t**≤* - Find the area under the curve for 2
*≤**t**≤*

**Show all work.**

- Given the polar equation r = 1 + cos (3
*θ*).- Sketch the curve.
- Find the area enclosed by the curve.

__2__*π*

- Set up the integral for the arc length of the curve with 0
*≤**θ**≤*

(d) Find the equation for the line tangent to the curve at *θ *=^{ }^{2}^{π}

**Show all work.**