#### To determine

**To find:**

The work needed to pump half of the water out of the aquarium

#### Answer

W=2450 J

#### Explanation

**1) Concept:**

Approximate the required work by using the concept of Riemann sum. Then express the work as an integral and evaluate it.

**2) Given:**

i) Length of aquarium 2m

ii) Width of aquarium 1m

iii) Depth of aquarium 1m

**3) Calculation:**

Let’s place the origin at the top of the aquarium and x-axis pointing downward as in the figure.

Divide the water into “slices” with the thickness of one slice ∆x.

If xi* is a point in the ith such interval, then all points in the interval are lifted approximately the same amount,xi*.

To pull half of the water out of the aquarium

Total water is 1m deep.

Therefore,0≤xi*≤12

The volume of each slice 2×1×∆xm3

By the formula,

Mass of each slice,

mass=density·volume

=(1000 kg/m3)(2∆x m3)

=2000∆x kg

By the formula,

f=mg

Where m is mass and g=9.8 m/s2 is acceleration due to gravity, so force on each slice

f=2000∆x kg(9.8 m/s2)

f=19,600∆x N

By the formula,

work=force·distance

=19,600∆x N·xi*

Total work done is

W=limn→∞∑i=1n19,600xi*∆x

=∫01219,600xdx

=9800x2012

=2450 J

**Conclusion:**

The work done to pump half of the water out of the aquarium is 2450 J