Use the shell method to find the volume of the solid generated by revolving the plane region about the line x = 9.
y = 7x − x2
y = 0
V = Int(2pi*r*h dx)[0,7]
where
r = 9-x
h = y
V = 2pi*Int((9-x)(7x-x^2)dx)[0,7]
= 2pi*Int(x^3 - 16x^2 + 63x)dx [0,7]
= 2pi(1/4 x^4 - 16/3 x^3 + 63/2 x^2)[0,7]
= 2pi(2401/4 - 5488/3 + 3087/2)
= 2pi * 3773/12
= 3773/6 pi
To find the volume of the solid generated by revolving the plane region about the line x = 9, we can use the shell method.
Step 1: Determine the boundaries of integration.
The boundaries of integration will be the x-values where the two curves intersect. Setting the two equations equal to each other and solving for x, we can find the points of intersection.
7x - x^2 = 0
x(7 - x) = 0
x = 0 or x = 7
Therefore, the boundaries of integration are x = 0 and x = 7.
Step 2: Set up the integral.
The volume can be calculated using the formula of the shell method:
V = 2π ∫[a, b] x · f(x) · dx,
where a and b are the boundaries of integration and f(x) represents the distance from the line of revolution to the curve at each x-value.
In this case, we are revolving about the line x = 9, so the distance from the line to the curve is 9 - x. Substituting this into the formula, we have:
V = 2π ∫[0, 7] x · (9 - x) · dx.
Step 3: Evaluate the integral.
Evaluating this integral will give us the volume of the solid.
V = 2π ∫[0, 7] (9x - x^2) · dx
= 2π [4.5x^2 - (1/3)x^3] |[0, 7]
= 2π [(4.5(7)^2 - (1/3)(7)^3) - (4.5(0)^2 - (1/3)(0)^3)]
= 2π [(4.5(49) - (1/3)(343)) - (0 - 0)]
= 2π [220.5 - 114.3333]
≈ 2π (106.1667)
≈ 667.034 cubic units (rounded to three decimal places).
Therefore, the volume of the solid generated by revolving the plane region about the line x = 9 is approximately 667.034 cubic units.
To find the volume of the solid generated by revolving the plane region about the line x = 9, we can use the shell method. The shell method is a technique in calculus used to find the volume of a solid of revolution.
Here's how we can proceed step by step using the shell method:
Step 1: Draw the graph of the given equations.
We have two equations: y = 7x - x^2 and y = 0. The region of interest is the area between the curve y = 7x - x^2 and the x-axis.
Step 2: Determine the limits of integration.
Since we are revolving the region about the line x = 9, we need to find the limits of integration for x. In this case, the region of interest lies between the x-values where the curve intersects the x-axis. To find those points, we set y = 0 in the equation y = 7x - x^2:
0 = 7x - x^2
x^2 - 7x = 0
x(x - 7) = 0
x = 0 or x = 7
So, the limits of integration will be from x = 0 to x = 7.
Step 3: Define the height of a typical shell.
The height of a typical shell is the distance between the line x = 9 and the curve at a particular x-value. In this case, the height will be 9 - x.
Step 4: Find the differential volume of a typical shell.
The differential volume of a typical shell is given by multiplying the height of the shell by the circumference of the shell and the thickness of the shell. The thickness of the shell is denoted by dx.
The circumference of the shell is given by 2πr, where r is the distance from the shell to the axis of rotation (in this case, x = 9).
Since the axis of rotation is x = 9, the distance from the shell to the axis of rotation is (9 - x).
So, the differential volume of a typical shell will be 2π(9 - x)(9 - x)dx.
Step 5: Integrate the differential volume to obtain the total volume.
We integrate the differential volume with respect to x over the limits of integration (from x = 0 to x = 7) to obtain the total volume:
V = ∫ [0 to 7] 2π(9 - x)(9 - x)dx
Evaluating this integral will give us the volume of the solid generated by revolving the plane region about the line x = 9.
Note: The integrals can be solved using integration techniques such as substitution or integration by parts.
Using these steps, you can find the volume of the solid generated by revolving the given plane region about the line x = 9 using the shell method.