The region bounded by the given curves is rotated about the specified axis. Find the volume V of the resulting solid by any method.
x = (y − 6)^2, x = 1; about y = 5
Luckily, the curve intersects x=1 at y=5 and y=7, so, using shells,
v = ∫[5,7] 2πrh dy
where r=y-5 and h=1-x=1-(y-6)^2
v = 2π∫[5,7] (y-5)(1-(y-6)^2) dy
= 2π(-y^4/4 + 17y^3/3 - 9y^2/2 + 175y) [5,7]
= 8π/3
To find the volume of the resulting solid, we can use the method of cylindrical shells. This involves finding the volume of a series of infinitely thin cylindrical shells and summing them up.
First, let's find the limits of integration. From the equation x = 1, we can see that y ranges from 5 to 7.
Now, let's determine the height of each cylindrical shell. The height of each shell will be the difference between the curves x = (y - 6)^2 and x = 1 at a given value of y. We can express this height as h = (y - 6)^2 - 1.
Next, we find the radius of each cylindrical shell. The radius will be the distance from the axis of rotation y = 5 to the curve x = (y - 6)^2. This distance can be expressed as r = y - 5.
The volume of each cylindrical shell is given by V = 2πrh, where r is the radius and h is the height. Therefore, V = 2π(y - 5)[(y - 6)^2 - 1].
Now, we integrate this expression with respect to y over the given limits of integration, which are y = 5 to y = 7, to find the total volume:
V = ∫[5 to 7] [2π(y - 5)[(y - 6)^2 - 1]dy]
Evaluating this integral will give us the volume V of the resulting solid.
To find the volume of the solid using the method of cylindrical shells, we need to integrate the circumference of the shell multiplied by its height, and then sum up these volumes for all shells.
The first step is to determine the limits of integration for the variable y.
For the given curves, we have x = (y - 6)^2 and x = 1.
Setting these two equations equal to each other, we have:
(y - 6)^2 = 1
Taking the square root of both sides, we get:
y - 6 = ±1
So, y - 6 = 1 or y - 6 = -1
This gives us two values for y:
y = 7 and y = 5.
Next, we need to express x in terms of y for the given curves.
We have x = (y - 6)^2.
Now, let's determine the radius and height of the cylindrical shell at a given value of y.
The radius is the distance from the axis of rotation (y = 5) to the curve x = (y - 6)^2.
So, the radius is given by r = x - 5.
Substituting x with (y - 6)^2, we have r = (y - 6)^2 - 5.
The height of the cylindrical shell is the difference between the upper and lower bounds of y, which is y = 7 - y = 5, so h = 2.
Finally, the volume of the cylindrical shell is given by V = 2πrh.
Substituting the expressions for h and r, we get:
V = 2π(2)((y - 6)^2 - 5).
To find the total volume, we need to integrate this expression over the range of y from 5 to 7:
V = ∫[5,7] 2π(2)((y - 6)^2 - 5) dy.
Evaluating this integral will give you the volume of the resulting solid.