Use the shell method to find the volume of the solid generated by revolving the plane region about the indicated line.
y=4x-x^2, y=0, about the line x=5
v = ∫[0,4] 2πrh dx
where r=5-x and h=y
= 2π∫[0,4] (5-x)(4x-x^2) dx
= 2π(x^4 - 3x^3 + 10x^2) [0,4]
= 64π
check using discs:
v = ∫[0,4] π(R^2-r^2) dy
where R=5-(2-√(4-y)) and r=5-(2+√(4-y))
= π∫[0,4] (3+√(4-y))^2-(3-√(4-y))^2
= π∫[0,4] (9+6√(4-y)+(4-y))-(9-6√(4-y)+(4-y)) dy
= π∫[0,4] 12√(4-y) dy
= 4π (-2 (4-y)^(3/2)) [0,4]
= 64π
To find the volume of the solid generated by revolving the plane region about the line x=5 using the shell method, follow these steps:
Step 1: Determine the limits of integration.
In this case, since the region is bounded by y=0 and y=4x-x^2, we need to find the values of y where the curves intersect. Setting the two equations equal to each other, we have:
0 = 4x - x^2
Rearranging and factoring, we get:
x^2 - 4x = 0
x(x - 4) = 0
x = 0 or x = 4
So, the limits of integration for x are from 0 to 4.
Step 2: Set up the integral for the volume using the shell method.
The shell method formula for finding volume is:
V = 2π ∫[a,b] (x * h(x)) dx
In this case, the height (h(x)) of each shell is the difference between the two curves: h(x) = 4x - x^2.
So, the integral for the volume becomes:
V = 2π ∫[0,4] (x * (4x - x^2)) dx
Step 3: Evaluate the integral.
To evaluate the integral, simplify the expression inside the integral and integrate term by term:
V = 2π ∫[0,4] (4x^2 - x^3) dx
V = 2π [ (4/3)x^3 - (1/4)x^4 ] |[0,4]
Evaluating the integral with the limits of integration, we get:
V = 2π [ (4/3)(4^3) - (1/4)(4^4) - (4/3)(0^3) + (1/4)(0^4) ]
V = 2π [ (4/3)(64) - (1/4)(256) ]
V = 2π [ 256/3 - 64 ]
V = 2π [ 128/3 ]
Step 4: Simplify the result, if needed.
Final volume = 256π/3
Therefore, the volume of the solid generated by revolving the plane region about the line x=5 is 256π/3 cubic units.
To find the volume of the solid generated by revolving the given region about the line x=5 using the shell method, we can follow these steps:
Step 1: Sketch the graph
First, let's sketch the given region y=4x-x^2 and the line x=5 in the coordinate plane to visualize the scenario. The region lies between the x-axis (y=0) and the graph of the equation.
Step 2: Identify the region of rotation
Next, identify the region of rotation. In this case, the region lies between the x-axis and the curve y=4x-x^2.
Step 3: Determine the limits of integration
To set up the integral for the volume, we need to determine the limits of integration with respect to x. Since we are revolving the region about the line x=5, the distance between the line of rotation and the curve will vary as we move along the x-axis.
To find the limits of integration, we need to find the x-values where the curve and the line intersect.
Setting the equation of the curve equal to the equation of the line:
4x - x^2 = 0
x(4 - x) = 0
From this equation, we can find that x = 0 and x = 4 are the x-values of the points where the curve intersects the line.
So, the limits of integration will be x = 0 to x = 4.
Step 4: Set up the integral
We will use the shell method for this problem. The volume of each shell can be found using the formula:
V = 2πrhΔx
Where r is the distance from the line of rotation (x=5) to the curve.
For the given problem, the distance r is equal to (5-x) since we are revolving the region about the line x=5.
Therefore, the volume element for each shell will be:
V = 2π(5-x)(4x-x^2)Δx
Step 5: Integrate
Now, we integrate the expression from x = 0 to x = 4 to find the total volume.
V = ∫[0 to 4] 2π(5-x)(4x-x^2) dx
After performing the integration, you will obtain the volume of the solid generated by revolving the region about the line x=5.
Note: Make sure to evaluate the integral properly or use appropriate software/calculator to obtain the numerical value of the volume.