A cylindrical water tank has radius r=1 meter and height 10 meters. Suppose the tank is lying on its side and is filled halfway with water. How much work does it take to pump all the water out the top of the tank? The height is now 2 meters.

Thanks, I'm not really sure how to cut into pieces to find the integral.

Whoops, Assume gravity is 9.8, wrong name

Each slice of water is a rectangle. Its length is 10. If the water has depth y, then the width of the rectangle is

√(1 - (1-y)^2) = √(2y-y^2)

Now you have to figure the water's depth (h), and the height to pump each slice of water is 2-y.

So, you have to integrate on y from 0 to h.

I'm sure google will reveal a similar problem.

To find the work required to pump all the water out of the tank, we need to calculate the amount of work done in lifting each infinitesimally thin slice of water from its initial height to the final height. Then we can integrate over the entire height of the tank to find the total work.

Let's start by considering an infinitesimally thin slice of water at a height h meters from the bottom of the tank. The volume of this slice can be approximated as the area of a circle (base of the slice) times the thickness of the slice. The area of the circle is given by πr^2, where r is the radius of the tank.

Since the tank is lying on its side, the slice will have thickness dh, and its volume can be approximated as πr^2 dh. We can express the radius in terms of the height, r = h/10, since the height of the tank is 10 meters.

Therefore, the volume of the slice is π(h/10)^2 dh. The weight of the slice of water is given by its volume multiplied by the density of water, which is approximately 1000 kg/m^3.

So, the weight of the slice is 1000π(h/10)^2 dh. To calculate the work done in lifting this slice from its initial height (h) to the final height (2 meters), we multiply the weight by the distance lifted, which is 2 meters - h.

Thus, the work done to lift this slice is given by dW = 1000π(h/10)^2 (2 - h) dh.

To find the total work, we integrate this expression over the height of the tank, from h = 0 (bottom) to h = 2 (top):

W = ∫[0 to 2] 1000π(h/10)^2 (2 - h) dh.

Evaluating this integral will give us the total work required to pump all the water out of the tank.

Note: To calculate the integral, you can simplify the expression inside the integral and use integration techniques such as substitution or integration by parts.