The area enclosed between the x-axis, the curve y=x(2-x) and the ordinates x=1 and x=2 is rotated through 2π radians about x-axis.
(a)Calculate the volume of the solid revolution formed.
(b)Calculate the rotating area.
from this question what about he grap need to draw or not
To answer this question, it is helpful to draw the graph of the curve y = x(2 - x) in order to visualize the shape and understand the region being rotated. The graph can be sketched by plotting points or by plotting the equation directly.
First, let's find the x and y-intercepts of the curve:
1. Set y equal to zero to find the x-intercepts:
x(2 - x) = 0
This equation can be satisfied when x = 0 or x = 2;
So, the x-intercepts are (0, 0) and (2, 0).
2. Set x equal to zero to find the y-intercept:
y = 0(2 - 0) = 0;
So, the y-intercept is (0, 0).
Now, we can plot these points on the xy-plane and connect them to form a curve that represents the equation y = x(2 - x). The curve will have a downward opening parabolic shape, passing through the points (0, 0), (1, 0), and (2, 0).
After visualizing the curve, we can proceed to calculate the volume of the solid and the rotating area.
(a) Calculating the volume of the solid revolution:
To find the volume of the solid formed, we can use the method of cylindrical shells. The volume of each shell is the circumference (2πr) multiplied by the height (y) of the shell. By summing up these volumes over the entire range of x values, we can find the total volume.
The height of the shell is given by y = x(2 - x), and the radius is the distance between the x-axis and the curve. Thus, the radius is equal to x.
Therefore, the volume of each shell is 2πx * (x(2 - x)) = 2πx^2(2 - x).
To find the total volume, integrate this expression over the interval x = 1 to x = 2:
V = ∫[1, 2] 2πx^2(2 - x) dx.
Evaluating this integral will give you the volume of the solid formed.
(b) Calculating the rotating area:
The rotating area is the area of the region enclosed between the curve y = x(2 - x), the x-axis, and the ordinates x = 1 and x = 2. It can be found by integrating the expression for the curve y = x(2 - x) over the same interval as before: [1, 2].
The rotating area can be obtained by evaluating the following integral:
A = ∫[1, 2] x(2 - x) dx.
By integrating this expression, you will obtain the exact value of the rotating area.