Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the y-axis.
y=4-x^2, y=0
The volume of a shell of radius r, height h and thickness dy (because revolving around the y-axis) is
2πrh dx
Now, our plane region is symmetric about the y-axis, so revolving all of it it is the same as revolving only the part for x>=0. o, we'll take the volume to be
∫[0,2] 2πrh dx
where r=x and h=y
= 2π∫[0,2] x(4-x^2) dx
= 2π(2x^2 - 1/4 x^4) [0,2]
= 8π
You can check using shells of thickness dy. The volume is then
∫[0,4] πr^2 dy
where r=x
= π∫[0,4] (4-y) dy
= π(4y - 1/2 y^2) [0,4]
= 8π
To set up and evaluate the integral using the shell method for finding the volume of the solid generated by revolving the plane region about the y-axis, follow these steps:
Step 1: Visualize the region and the resulting solid.
Graph the given equation y = 4 - x^2 and the line y = 0 to identify the region that will be revolved around the y-axis. The resulting solid will be a three-dimensional shape with a cylindrical-like form.
Step 2: Determine the limits of integration.
To find the limits of integration, determine the points of intersection between the curves y = 4 - x^2 and y = 0. Set both equations equal to each other and solve for x:
4 - x^2 = 0
Rearrange the equation:
x^2 = 4
Take the square root:
x = ±2
So the limits of integration will be x = -2 to x = 2.
Step 3: Establish the radius of the cylindrical shells.
The radius of each cylindrical shell is the perpendicular distance from the y-axis to the function y = 4 - x^2. Since the axis of rotation is the y-axis, the radius will simply be the value of x.
Therefore, the radius, r, is equal to x.
Step 4: Determine the height (width) of each cylindrical shell.
The height (width) of each cylindrical shell is the difference in y-values between the two curves, y = 4 - x^2 and y = 0. It can be expressed as the function h:
h = (4 - x^2) - 0
= 4 - x^2
Step 5: Set up the integral.
The volume of each cylindrical shell, dV, is given by the formula:
dV = 2πrhdx
Where 2π represents the circumference of the cylindrical shell.
To find the total volume, integrate the equation with respect to x, using the limits of integration determined previously:
V = ∫[from -2 to 2] 2π(x)(4 - x^2) dx
Step 6: Evaluate the integral.
Integrate the equation using the limits of integration:
V = 2π ∫[from -2 to 2] (4x - x^3) dx
After integrating, you should obtain the final answer for the volume of the solid generated by revolving the region about the y-axis.