if the region enclosed by the y-axis, the line y=2 and the curve y=the square root of x is revolved about the y-axis, the volume of the solid generated is????
To find the volume of the solid generated by revolving the region enclosed by the y-axis, the line y=2, and the curve y = √x about the y-axis, we can use the method of cylindrical shells.
First, let's sketch the region to get a better understanding. The region enclosed by the y-axis, the line y=2, and the curve y=√x looks like a right triangle, where the base is along the y-axis, the height is along the line y=2, and the curved side is along the graph of y=√x.
Since we are revolving this region about the y-axis, the cylindrical shells will be perpendicular to the y-axis.
Now, let's set up the volume integral using cylindrical shells:
The radius of each cylindrical shell is the x-coordinate of the curve at that particular height, which is √x.
The height of each cylindrical shell is the difference between y=2 and y = √x, which is (2 - √x).
The thickness of each cylindrical shell is an infinitesimally small change in y, which we represent as dy.
Therefore, the volume of each cylindrical shell is given by dV = 2π√x (2 - √x) dy.
To find the total volume of the solid, we need to integrate this expression with respect to y, over the range where y varies from 0 to 2 (since the line y=2 is the upper boundary of the region):
V = ∫[0 to 2] 2π√x (2 - √x) dy.
To evaluate this integral, we need to express everything in terms of y instead of x. Rearranging the equation y = √x, we get x = y^2.
Now, we can substitute x = y^2 into the expression and rewrite the integral as:
V = ∫[0 to 2] 2π√(y^2) (2 - √(y^2)) dy.
Simplifying further, we have:
V = ∫[0 to 2] 2πy (2 - |y|) dy.
We use the absolute value |y| to account for the fact that the expression inside the square root can be positive or negative, depending on the value of y.
Finally, integrating this expression will give us the volume of the solid generated:
V = ∫[0 to 2] 2πy (2 - |y|) dy.
Evaluating this integral will give you the volume of the solid generated.