Please Show steps as I would like to understand the material... a tank of revolu

Question

Please Show steps as I would like to understand the material...

a tank of revolution Suppose a tank is created by revolving the region R bounded by the graphs of x = g(y), x = 0, y = 0, and y = H about the y-axis, where g(y) > 0 on [0, H] (Figure 7). Show that if the tank is filled to depth h, then the volume of water in the tank is V(h) = pi g(y)2 dy. Find the volume function V=f(h), for the tank formed when the region R bounded by the graphs of y = x + 1, x = 0, y = 0, and y = 4 is revolved about the y-axis. Find the volume function V=f[h), for the tank formed when the region R bounded by the graphs of x = y2 + l, x = 0 , y =0, and y = 3is revolved about the y-axis.

Explanation / Answer

As curve g(h) revolves it forms a tank like structure whose radius increases ,that is radius=f(g(h))

it is a function of g(h)

Volume=Area*height

Area=pi*r^2

r=g(h)

h=y

dh=dy

then total volume=pi*integral[g(h)^2]*dy limits are from 0 to h.

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