A semicircular plate with radius 8 m is submerged vertically in water so that the top is 3 m above the surface. Express the hydrostatic force against one side of the plate as an integral and evaluate it. (Round your answer to the nearest whole number. Use 9.8 m/s2 for the acceleration due to gravity. Recall that the weight density of water is 1000 kg/m3.)

If the square end is under water: 312271.92  N

If the round end is under water:   262976.67 N

Step-by-step explanation:

We have to:

P = ρ·g·d

where:

P is hydrostatic pressure, g is gravitational acceleration, d is depth, ρ is fluid density

Force = hydrostatic pressure x submerged area

F = PA

You didn't specify the orientation

If the square end is under water:

F = (1000 kg/m³) (9.81 m/s²) (2) ∫ (2-y)√(64 - y²) dy from 0 to 2

F = 19620 ∫ (2-y)√(64 - y²) dy from 0 to 2

The result of the solution of the integral is 15.916

Thus:

F = 15.916*19620 = 312271.92  N

If the round end is under water:

F = (1000 kg/m³) (9.81 m/s²) (2) ∫ (y - 3) √(64- y²) dy from 3 to 5

F = 19620 ∫ (y - 3) √(25 - y²) dy from 3 to 5

The result of the solution of the integral is 13.403

Thus:

= 13.403*19620 = 262976.67 N

A = pi * r^2

 = pi * (2x^-3)^2

 = pi * 2^2 * (x^-3)^2

 = pi * 4 * (x^-6)

 = 4*pi * x^-6