How would I approach this question - shell method or washer method to integrate?
Consider a spherical fishbowl with a radius of 10 cm. The top of the fishbowl is cut-off at a height of 15 cm to allow the bowl to be filled. Find the amount of water it would take to fill the fishbowl to the top.
Assume the center of the bowl is placed on the origin of coordinate system, i.e. (0,0).
Consider the fish bowl to be made up of very thin cylinders, of thickness, `Deltay`. The radius of the circles is expressed in terms of variable x and the thickness in terms of variable y. The volume of such a thin cylinder would be `Pi*x^2*Deltay`
` ` ` `
Thus the volume of the fish bowl would be the sum of all such cylinders, for y= -10 to y=5. This can be calculate by integrating the function for volume of a cylinder, ` Pi*x^2*Deltay` over the interval [-10, 5].
In the circle (base of the cylinder), `x^2+y^2 = r^2 ==> x^2 = r^2 - y^2`
` `Therefore the area can be written as `Pi*(r^2-y2)*Deltay`
Therefore the volume of the fish bowl would be the sum of all such cylinders. This can be calcualted by integrating the area of teh cylinder over the range of values of y. Since we are assuming the center of the sphere is in the center of the coordiante system, the range would be from -10,5
Thus the volume is = `int_-10^5 *(pi*(100-y^2)) dy `
` ``= pi* [100y - y^3/3]_-10^5`
`= pi* (100*5-5^3/3) - (100*-10 - (-10)^3/3)`
` ` `= pi*1125 = 3534.29 cm^3 = 3.53 l`
Volume of water required to fill the fish bowl is 3.53 l