Dear All
Can you please let me know how to calculate the volume obtained by rotating the y=f(x) curve about the x-axis for x=x1 to x2?
Can you please let me have the answer for rotating it about the y-axis for y=y1 to y2 as well ?
Thank you
volume?
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freeabelian
- Posts: 9
- Joined: Sat Jan 01, 2011 5:35 pm
Re: volume?
The easiest way to do this is through the washer method.
$$V = \pi\int_{x_1}^{x_2}(f(x)^2)dx$$
This works for rotation along the x-axis. Intuitively, you're adding up loads of infinitesimally small washers of area $$\pi f(x)^2.$$
If the function is being rotated around the y axis, you'll need to get the curve as a function of y instead, as we'll have a dy integral, and then apply the exact same procedure.
alternatively, you can use the cylinder method.
$$V=2\pi\int_{x_1}^{x_2}x |{f(x)}| dx$$
$$V = \pi\int_{x_1}^{x_2}(f(x)^2)dx$$
This works for rotation along the x-axis. Intuitively, you're adding up loads of infinitesimally small washers of area $$\pi f(x)^2.$$
If the function is being rotated around the y axis, you'll need to get the curve as a function of y instead, as we'll have a dy integral, and then apply the exact same procedure.
alternatively, you can use the cylinder method.
$$V=2\pi\int_{x_1}^{x_2}x |{f(x)}| dx$$