Find the volume of the solid formed by revolving the region bounded by the graphs of y=4x^2 and y=16 about the x-axis. (Washer Volume)
integral from y = 0 to y = 16 (origin to x = 2)
find volume of outer cylinder of radius 16
Vol - pi r^2 h = pi (16^2) (2)
find volume generated by inner curve between x axis and y = 4 x^2
Vinner = pi y^2 dx from 0 to 2
= pi 16 * integral x^4 dx
= 16 pi (32/5)
then subtract the inner from the outer
2 pi (16^2) - 16 pi (32/5)
= 32 pi *16 - 32 pi *16/5
= 32 pi (4/5) (16)
Oh, double that to include the left side.
To find the volume of the solid formed by revolving the region bounded by the graphs of y=4x^2 and y=16 about the x-axis using the washer method, we can follow these steps:
Step 1: Find the points of intersection of the two curves.
Setting the two equations equal to each other and solving for x:
4x^2 = 16
Divide both sides by 4:
x^2 = 4
Taking the square root of both sides:
x = ±2
The region bounded by the two curves lies between x = -2 and x = 2.
Step 2: Set up the integral for the volume.
The volume of a washer is given by the formula:
V = ∫[a,b] π(R^2 - r^2) dx
where a and b are the limits of integration, R is the outer radius, and r is the inner radius.
In this case, the outer radius (R) is the distance from the axis of revolution (x-axis) to the curve y = 16, and the inner radius (r) is the distance from the axis of revolution to the curve y = 4x^2.
Outer radius (R): y = 16
Inner radius (r): y = 4x^2
Thus, the integral will be:
V = ∫[-2, 2] π((16)^2 - (4x^2)^2) dx
Step 3: Calculate the integral.
Simplifying the integral:
V = ∫[-2, 2] π(256 - 16x^4) dx
Integrating:
V = π ∫[-2, 2] (256 - 16x^4) dx
V = π [256x - (16/5)x^5] |[-2, 2]
V = π [(256(2) - (16/5)(2)^5) - (256(-2) - (16/5)(-2)^5)]
V = π [512 - (16/5)(32) + 512 + (16/5)(32)]
V = π [1024]
Step 4: Finalize the answer.
The volume of the solid formed by revolving the region bounded by the graphs of y=4x^2 and y=16 about the x-axis using the washer method is 1024π, where π is approximately equal to 3.14.
Therefore, the volume is 3203.84 cubic units.
To find the volume of the solid formed by revolving the region bounded by the graphs of y = 4x^2 and y = 16 about the x-axis, we can use the method of washers.
First, let's visualize the region bounded by the graphs:
The graph of y = 4x^2 is a parabola that opens upward and intersects the x-axis at (0, 0). The graph of y = 16 is a horizontal line that intersects the y-axis at (0, 16). The two graphs intersect at x = ±2.
To calculate the volume using the washer method, we'll integrate over the region of interest and sum up the volumes of infinitesimally thin washers.
The radii of the washers will vary as we move along the x-axis. At each value of x, the radius is given by the difference between the two curves. Therefore, the radius (R) of each washer is R = 16 - 4x^2.
The thickness (dx) of each washer is an infinitesimally small change in x.
The volume (dV) of each washer is calculated as follows:
dV = π(R_outer^2 - R_inner^2)dx
In this case, since we are revolving about the x-axis, the R_outer is 16 - 4x^2, and the R_inner is 0.
Now we can find the total volume by integrating the volume of each washer over the region:
V = ∫[a,b] π(R_outer^2 - R_inner^2)dx
In this case, the region is symmetric about the y-axis, so we can integrate from x = -2 to x = 2:
V = ∫[-2,2] π((16 - 4x^2)^2 - 0)dx
To proceed with the integration, we need to expand and simplify the expression inside the integral, and then integrate term by term:
V = π∫[-2,2] (256 - 128x^2 + 16x^4)dx
Integrated term by term, we get:
V = π(256x - 128/3x^3 + 16/5x^5) |[-2,2]
Evaluating the definite integral, we have:
V = π(512 - 128/3(8) + 16/5(32) - (-512 + 128/3(-8) + 16/5(-32)))
Finally, simplifying the expression, we can find the numerical value of the volume.