Find the Volume V of the solid of revolution generated by revolving the region bounded by the x- axis and the graph of y=4x-x^2 about the line y=4
Using discs of thickness dx, we get
v = ∫[0,4] πr^2 dx
where r=y = 4x-x^2
v = ∫[0,4] π(4x-x^2)^2 dx = 512π/15
Using shells of thickness dy, and exploiting the symmetry of the region (as we could have done above, also)
v = 2∫[0,4] 2πrh dy
where r = y and h = 2-x = 2-(2-√(4-y)) = √(4-y)
v = 2∫[0,4] 2πy√(4-y) dy = 512π/15
To find the volume of the solid of revolution, we can use the method of cylindrical shells.
Step 1: Identify the limits of integration.
Since we are rotating around the line y=4, we need to find the x-values where the curve intersects the line y=4. So let's set the equation of the curve equal to 4 and solve for x:
4x - x^2 = 4
Rearranging the equation:
x^2 - 4x + 4 = 0
Factoring the quadratic equation:
(x - 2)^2 = 0
Solving for x:
x - 2 = 0
x = 2
So, the limits of integration for x are 0 to 2.
Step 2: Determine the height of the cylindrical shell.
The height of each cylindrical shell is given by the difference in the y-values between the curve and the line of rotation. In this case, the line y=4. So, the height of the cylindrical shell is given by:
h = 4x - x^2 - 4
Step 3: Calculate the circumference of each cylindrical shell.
The circumference of each cylindrical shell is given by 2πr, where r is the x-value. In this case, r is the distance from the x-axis to the curve. So, r = x.
Step 4: Integrate the cylindrical shells.
The volume of each cylindrical shell is given by the product of its circumference and height: V_shell = 2πrh.
The total volume is the integral of the volume of each cylindrical shell from the lower limit to the upper limit:
V = ∫[lower limit to upper limit] (2πrh) dx
V = ∫[0 to 2] (2πx(4x - x^2 - 4)) dx
Now we can integrate the expression to find the volume of the solid.
To find the volume V of the solid of revolution generated by revolving the region bounded by the x-axis and the graph of y=4x-x^2 about the line y=4, we can use the method of cylindrical shells.
First, let's sketch the region bounded by the x-axis and the graph of y=4x-x^2. This region is a parabolic shape that opens downwards.
To find the limits of integration, we need to find the x-values where the parabola intersects the x-axis. Setting y=0 in the equation y=4x-x^2, we have:
0 = 4x - x^2
x^2 - 4x = 0
x(x - 4) = 0
So, we have x=0 and x=4 as the limits of integration.
Now, let's consider a cylindrical shell of thickness Δx and height y=4. The radius of this shell is given by the distance from the line y=4 to the parabola y=4x-x^2.
The distance between the line y=4 and the parabola can be found by subtracting the equation of the line from the equation of the parabola:
(4x - x^2) - 4 = 4x - x^2 - 4 = -x^2 + 4x - 4
The radius of the cylindrical shell at a given x-value is therefore (-x^2 + 4x - 4).
The volume V of each cylindrical shell is given by V = 2πrhΔx, where r is the radius, h is the height, and Δx is the thickness.
Now, we can write the integral for the volume V as follows:
V = ∫(2π)(-x^2 + 4x - 4)(4) dx
V = 8π∫(-x^2 + 4x - 4) dx
V = 8π[-(1/3)x^3 + 2x^2 - 4x] evaluated from x=0 to x=4
Evaluating the integral with the limits of integration, we get:
V = 8π[-(1/3)(4^3) + 2(4^2) - 4(4)] - 8π[-(1/3)(0^3) + 2(0^2) - 4(0)]
V = 8π(-64/3 + 32) - 8π(0)
V = 8π(-64/3 + 96)
V = 8π(32/3)
V = 256π/3
Therefore, the volume of the solid of revolution is V = 256π/3 cubic units.