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Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the y-axis. y=18x,y=18, and x=0 (Type exact answers.) A. ∫ dy B. ∫1dx

User Ben Hall
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Given curves are y=18x, y=18, and x=0. We have to find the volume of the solid generated when R is revolved about the y-axis using the shell method.Let's sketch the curves first:We can see that R is the region bounded by the curves y=18x, y=18 and x=0.

We need to rotate this region around the y-axis using the shell method to find the volume of the solid generated.Let's use the shell method here:Shell method:$$\large V=2\pi \int_{a}^{b}x[f(x)-g(x)]dx$$

Here, we need to rotate R around the y-axis, therefore, a = 0 and b = 18.

From the graph, we can see that the functions $f(x)$ and $g(x)$ are as follows:$$f(x) = 18 \\ g(x) = 18x$$

Using the shell method, the volume of the solid generated is:$$\begin{aligned}V &= 2\pi \int_{0}^{18}x[18-18x]dx \\&= Therefore, the volume of the solid generated when R is revolved about the y-axis is 17496π.

User Denis Balko
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