A rectangular sea aquarium observation window is 12 ft wide

Question

Solve the problem.

1) Find a curve through the point (0, 3) whose length integral, 0 ≤ x ≤ 1, is L =

₀∫¹√(1+4x²) dx.

Set up an integral for the area of the surface generated by revolving the given curve about the indicated axis.

2) xy = 3, 1 ≤ y ≤ 2; y-axis

Solve the problem.

3) A vertical right circular cylindrical tank measures 24 ft high and 12 ft in diameter. It is full of oil weighing 60 lb/ft³. How much work does it take to pump the oil to the level of the top of the tank? Give your answer to the nearest ft · lb.

4) A rectangular sea aquarium observation window is 12 ft wide and 4 ft high. What is the force on this window if the upper edge is 3 ft below the surface of the water? The density of seawater is 64 lb/ft³.

Find the fluid force exerted against the vertically submerged flat surface depicted in the diagram. Assume arbitrary units, and call the weight-density of the fluid w.

5)

Find the center of mass of a thin plate covering the given region with the given density function.

6) The region enclosed by the parabolas y = 50 - x² and y = x², with density δ(x) = x²

Find the centroid of the thin plate bounded by the graphs of the given functions. Use δ = 1 and M = area of the region covered by the plate.

7) g(x) = x² and f(x) = x + 12

Find the length of the curve.

8) y = (16 - x^2/3)^3/2 from x = 1 to x = 64

Find the area of the surface generated by revolving the curve about the indicated axis.

9) x = 3√(4 - y), 0 ≤ y ≤ 15/4; y-axis

Find the mass of a thin plate of constant density covering the given region.

10) The region between the x-axis and the curve y = 2csc²x, π/4 ≤ x ≤ 3π/4.