I’m studying and need help with a Calculus question to help me learn.
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:
x = sin θ,
y = cos θ,
−π ≤ θ ≤ π
7. Let x = √2 t2 + 6, y = 1 t3 – 1 t.
2 3 2
- Find dy/dx.
- Find d2y/dx2.
- Find the length of the curve for −1 ≤ t ≤ 1.
- Find the area under the curve for 2 ≤ t ≤ 3
Show all work.
- Given the polar equation r = 1 + cos (3θ).
- Sketch the curve.
- Find the area enclosed by the curve.
- 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.