Find the volume of the solid generated by revolving the region bounded by the line y=5x+6 and the parabola y=x^2 about the following lines:
a) the line x=6
b) the line x=-1
c)the x axis
d) the line y=36
Thanks!
the curves intersect at (-1,1) and (6,36)
The x-line rotations all work the same way. Shells probably work best:
v = ∫[-1,6] 2pi * r * h dx
where r is 6-x or 1+x and h = (5x+6)-x^2
simple polynomial
for the y-line axes, maybe discs work best, but you'll have to use washers, since the discs will have holes in them.
v = ∫[-1,6] pi * (R^2-r^2) dx
where R=(y-6)/5 and r = x^2
again simple polynomials to integrate.
Come on back if you get stuck.
To find the volume of the solid generated by revolving a region about a line, we can use the method of cylindrical shells or the method of disks/washers, depending on the problem.
a) Revolving about the line x = 6:
In this case, we will use the method of cylindrical shells.
First, let's find the points of intersection between the line y = 5x + 6 and the parabola y = x^2.
Set them equal to each other:
5x + 6 = x^2
Rearrange the equation to get:
x^2 - 5x - 6 = 0
Factorizing the equation, we have:
(x - 6)(x + 1) = 0
So, x = 6 or x = -1.
To determine the limits of integration, we look for the extreme values of x for the region.
Since x = -1 is on the left and x = 6 is on the right, the limits of integration will be -1 and 6.
The radius of each cylindrical shell will be the distance from the line x = 6 to the x-coordinate of each point on the parabola.
The height of each cylindrical shell will be the difference between the line y = 5x + 6 and the parabola y = x^2.
The volume of each cylindrical shell can be calculated using the formula:
V = 2πrhΔx
where r is the radius, h is the height, and Δx is the differential length along the x-axis.
Integrating the expression will give us the total volume:
V = ∫(2πrh)dx from -1 to 6.
b) Revolving about the line x = -1:
Follow the same steps as in part a), but this time, the limits of integration will be different. You will integrate from -1 to 6 again since x = -1 is now on the right and x = 6 is on the left.
c) Revolving about the x-axis:
In this case, we will use the method of disks/washers.
Since the region is being revolved around the x-axis, the radius of each disk will be the y-coordinate of each point on the line y = 5x + 6 or the parabola y = x^2, whichever is greater.
The height of each disk will be the difference between the y-coordinate of the parabola and the line.
The formula for the volume of each disk is:
V = πr^2hΔx
To find the volume of the solid, integrate the expression:
V = ∫(πr^2h)dx from the x-coordinate of the leftmost point of intersection to the x-coordinate of the rightmost point of intersection.
d) Revolving about the line y = 36:
Similar to part a), find the points of intersection between the line y = 5x + 6 and the parabola y = x^2.
However, this time, the radius of each cylindrical shell will be the distance between the y-coordinate of each point on the parabola and the line y = 36. The height of each cylindrical shell will be the difference between the line y = 36 and the parabola y = x^2.
The volume of each cylindrical shell is given by the formula:
V = 2πrhΔx
Integrate the expression to get the total volume:
V = ∫(2πrh)dx from the x-coordinate of the leftmost point of intersection to the x-coordinate of the rightmost point of intersection.
Remember to calculate the integrals and plug in the values to obtain the final volume for each part (a), b), c), and d).