show how to approximate the required work by a Riemann sum.

[lil_hawk]

I'm nearly deaf and so my teacher's lectures don't always help and neither does the textbook which I've read most of. I need help setting up the integral because right now all I really know I think is that I need find the area of the cross section.

Instructions: Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it:

A circular swimming pool has a diameter of 24 ft, the sides are 5 ft high, and the depth of the water is 4 ft. How much work is required to pump all of the water out over the side? (Use the fact that water weighs 62.5lb/ft^3.)

Any help is greatly appreciated.

 

[hypoid]

Let dA – elementary work, then dA=g*dm*h, g=9,8 m/s2, dm – mass of elementary volume dV (dV=(p*D^2*dh)/4, dh – elementary height of water layer).
dm=rdV, r=62.5 lb/ft^3.
So dA=(g*r*p*D^2*dh)*h, h varies from 1 to 5 feet.
A=Int(1,5){(g*r*p*D^2*dh)*h}
A=(r*p*D^2*(h1^2- h2^2))/8-, where h1=5ft, h2=1ft.

 

[galactus]

Imagine the water being divided into n thin layers with thicknesses:

\(\displaystyle {\Delta}x_{1}, {\Delta}x_{2}, {\Delta}x_{3}, .....,{\Delta}x_{n}\)

The force needed to move the kth layer equals the weight of the layer, which can be found by multipltying its volume by 62.5.

Since the kth layer is a cylinder of radius r=12 and height \(\displaystyle {\Delta}x_{k}\), the force needed is:

\(\displaystyle \L\\({\pi}(12)^{2}{\Delta}x_{k})(62.5)=9000{\pi}{\Delta}x_{k}\)

The kth layer has a finite thickness and the upper and lower surfaces are at different distances from the origin. If the layer is thin, the difference in these distances is small, and we can say the entire layer is a single distance \(\displaystyle x_{k}\) from the origin.

Therefore, the work needed to pump all n layers will be:

\(\displaystyle \L\\W=\sum_{k=1}^{n}(5-x_{k})(9000{\pi}){\Delta}_{k}\)

Take the limit as \(\displaystyle max{\Delta}x_{k}\rightarrow{0}\).

\(\displaystyle \L\\9000{\pi}\int_{0}^{4}(5-x)dx\)

 

[lil_hawk]

Thanks. I'm going to need this for my test in a couple days.