**Calculus**

Manmohan Dash, g6pontiac@gmail.com

An aquarium 2 m long, 1 m wide, and 1 m deep is full of water. Find

(a) The hydrostatic pressure on the bottom of the aquarium,

(b) The hydrostatic force on the bottom, and the hydrostatic force on one end of the aquarium.

Answer;

(a) The hydrostatic pressure on the bottom is obviously *ρgh* where *ρ* = 1000*kg/m*^{3} and *g* = 10*ms*^{2}. Also *h* = 1*m*. So *P* = *ρgh* = 1000 × 10 × 1 = 10000 *Pa*. *Pa* = *Pascals*; SI unit of pressure.

(b) Now the force on bottom is *F* = *P* × *A* = 10000 × 2 × 1 = 20000 *N*, where SI unit of force is Newton (*N*).

For the force on the end of the aquarium we need to take each layer of side wall, as the pressure varies with depth. So *F* = *P* × *A* = ∫^{1}_{0}*ρ*.*g*.*a*.*xdx* where *a* is the width of the side wall and *x* is the height of the side wall from bottom (so *a* × *dx* = *A*, area of the strip).

The limit of integration begins from depth of the aquarium to the height of the same, hence (0, 1) Since *a* = 1, *F* = 10000 × ∫^{1}_{0}*xdx* = 1/2 × 10000 = 5000 *N*.

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