The region in the first quadrant enclosed by the graphs of #y=x# and #y=2sinx# is revolved about the x-axis, how do you find the volume of the solid generated?

Question

William Abdalla · Accepted Answer

See the answer below:

Adam Adkins · Answer

undefined To find the volume of the solid generated by revolving the region enclosed by the graphs of $ y = x $ and $ y = 2\sin(x) $ about the x-axis, you can use the method of cylindrical shells. The formula to find the volume using cylindrical shells is:

$$ V = \int_{a}^{b} 2\pi x \cdot (f(x) - g(x)) \, dx $$

Where:
- $ a $ and $ b $ are the x-values where the graphs intersect,
- $ f(x) $ is the upper function (in this case, $ f(x) = 2\sin(x) $),
- $ g(x) $ is the lower function (in this case, $ g(x) = x $).

First, find the points of intersection of the two curves by solving $ x = 2\sin(x) $ and $ x = x $. Once you have the intersection points, set up the integral using the formula above and integrate with respect to $ x $ from the lower x-value to the upper x-value.

After integrating, you'll have the volume of the solid generated by revolving the region about the x-axis.