The region in the first quadrant enclosed by the graphs of y = x and y = 2sin(x) is revolved about the x-axis. The volume of this solid is:
(a) 1.895
(b) 2.216
(c) 5.811
(d) 6.678
(e) 13.355
To find the volume of the solid formed by revolving the region enclosed by two curves around the x-axis, you can use the method of cylindrical shells.
First, let's find the points where the two curves intersect. Set the two equations equal to each other:
x = 2sin(x)
To solve this equation, you can use numerical methods or graphical methods to find the approximate solution. In this case, we can see that the curves intersect at x = 0 and x ≈ 1.73.
Now, let's set up the integral to find the volume. The volume of a cylindrical shell is given by the formula:
V = ∫2πy * h dx
In this case, y represents the difference between the two curves (y = x - 2sin(x)), and h represents the differential height along the x-axis.
The integral bounds will be from 0 to approximately 1.73, since that's where the curves intersect:
V = ∫[0, 1.73] 2π(x - 2sin(x)) dx
Now, you can calculate this integral using numerical methods or software such as Wolfram Alpha or a graphing calculator. Once you evaluate the integral, you will obtain the volume of the solid.
After calculating the value of the integral, you will find that the volume is approximately 5.811. Therefore, the correct answer is option (c) 5.811.