Set up, but do not evaluate, a definite integral representing the volume of the
solid formed by revolving the region of the plane bounded by the curves y = e
x
, x = 0 and
y = e about the line x = 1 using the disk/washer method.
you have washers of thickness dy, so
v = ∫[1,e] π(R^2 - r^2) dy
where R = 1 and r = 1-x = 1-lny
v = ∫[1,e] π(1^2 - (1 - lny)^2) dy
just as a check, using shells of thickness dx,
v = ∫[0,1] 2πrh dx
where r = 1-x and h = e-y = e-e^x
v = ∫[0,1] 2π(1-x)(e-e^x) dx
To set up the definite integral representing the volume of the solid, we will use the disk/washer method.
First, let's sketch the region in the xy-plane. We have the curves y = e^x, x = 0, and y = e.
The region bounded by these curves looks like a triangle with the base on the x-axis and the two curved sides.
To revolve this region about the line x = 1, we will consider a small vertical strip of width dx at a certain x-value.
When we revolve this strip about the line x = 1, it forms a disk with radius r and thickness dx.
The radius of the disk is the distance from the curve y = e^x to the axis of rotation (x = 1).
The distance from the curve y = e^x to the line x = 1 is given by r = 1 - x.
Now we need to express the height (thickness) of the disk. This is given by the difference in the y-values of the curves at the x-value.
The upper curve is y = e, and the lower curve is y = e^x. So the height of the disk is h = e - e^x.
Now, we can set up the integral to find the volume:
∫[from 0 to 1] π(r^2 - R^2) dx,
where r = 1 - x (the inner radius) and R = e - e^x (the outer radius).
Plugging in these values, we have:
∫[from 0 to 1] π((1 - x)^2 - (e - e^x)^2) dx.
So, the setup of the definite integral representing the volume of the solid is:
∫[from 0 to 1] π((1 - x)^2 - (e - e^x)^2) dx.
To set up a definite integral representing the volume of the solid formed by revolving the region of the plane bounded by the curves y = e^x, x = 0, and y = e about the line x = 1 using the disk/washer method, we will follow these steps:
1. Determine the limits of integration:
The region is bounded by the curves y = e^x, x = 0, and y = e. To find the limits of integration, we need to determine the x-values where these curves intersect.
First, set y = e^x equal to y = e and solve for x:
e^x = e
Taking the natural logarithm of both sides:
x = ln(e)
Since ln(e) = 1, the curves intersect at x = 1.
Therefore, the limits of integration for x will be from 0 to 1.
2. Determine the radius of each disk/washer:
The radius of each disk/washer will be the distance between the line x = 1 and the corresponding x-value in the region.
Since we are revolving the region about the line x = 1, the radius will be 1 - x.
3. Determine the height of each disk/washer:
The height of each disk/washer will be the difference between the upper and lower curves at a specific x-value.
The lower curve is y = e^x and the upper curve is y = e, so the height will be (e^x) - e.
4. Set up the integral:
The volume of each disk/washer can be approximated by the formula for the volume of a cylinder: V = πr^2h, where r is the radius and h is the height.
Since we are using the disk/washer method, the integral will be:
∫[from x = 0 to x = 1] π(1 - x)^2((e^x) - e) dx
Note that we square the radius (1 - x) since it represents the distance from x = 1.
Remember, this is just the setup for the definite integral. To evaluate the integral and get the actual volume, you would need to solve it numerically or symbolically.