Calculus 2

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 − x2 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/m³ for the density of the seawater in the Atlantic ocean, and 9.81m/s² for the force of gravity.

4. (a) Find the centroid of the region bounded by y = ¹ x + ¹ , y = x³, 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 = x³, 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 t² + 6, y = ¹ t³ – ¹ t.

2 3 2

1. Find dy/dx.
2. Find d²y/dx².

1. Find the length of the curve for −1 ≤ t ≤ 1.
2. Find the area under the curve for 2 ≤ t ≤ 3

Show all work.

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

2π

1. 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 = ^2π

Show all work.