Use the disk method to find the volume of the solid generated when the region bounded by 4= 1/(fourth root of (1-11x)), y=0, x=0, and x=1/22.
Int (1-11x)^(-1/4) dx [0,1/22]
= (1-11x)^3/4 * 4/3 * -1/22 [0,1/22]
= -2/33 (2^1/4 - 2)
That should be (1-11x)^3/4 * 4/3 * -1/11
To find the volume of the solid generated by the given region using the disk method, we need to integrate the area of the disks perpendicular to the x-axis.
First, let's start by finding the limits of integration.
The given region is bounded by the following curves:
- The x-axis, y = 0
- The vertical lines, x = 0 and x = 1/22
- The function, 4 = 1/(fourth root of (1-11x))
To find the limits of integration, we need to solve for the x-values where the curves intersect.
1. First, let's consider the curve defined by 4 = 1/(4√(1-11x)).
Rearrange the equation to get rid of the fraction:
4 = 4√(1-11x)
2. Square both sides to eliminate the radical:
4^2 = (4√(1-11x))^2
16 = 4(1-11x)
3. Simplify:
16 = 4 - 44x
44x = 4 - 16
44x = -12
x = -12/44
x = -3/11
So, the curve defined by 4 = 1/(4√(1-11x)) intersects the x-axis at x = -3/11.
Now let's consider the vertical lines:
- x = 0
- x = 1/22
Now we have our limits of integration: from x = -3/11 to x = 1/22.
Next, we need to set up the integral for finding the volume using the disk method. The volume element in the disk method can be represented by dV = π(r(x))^2dx, where r(x) is the radius of the disk at a particular x-value.
To calculate the radius, we need to express the given curve in terms of x. Let's rearrange the equation 4 = 1/(fourth root of (1-11x)).
1. Start by isolating the fraction:
4 (fourth root of (1-11x)) = 1
2. Raise both sides to the power of 4 to get rid of the fourth root:
(4 (fourth root of (1-11x)))^4 = 1^4
256 (1-11x) = 1
3. Solve for (1-11x):
256 (1-11x) = 1
1 - 11x = 1/256
11x = 1 - 1/256
11x = (256 - 1)/256
11x = 255/256
x = (255/256)/11
x = 255/2816
So, the equation 4 = 1/(fourth root of (1-11x)) can be rewritten as x = 255/2816.
Now, the radius of the disk at any given x-value is given by r(x) = x - 0 = x.
The integral for finding the volume is:
V = ∫[a,b] π(r(x))^2 dx
= ∫[(−3/11),(1/22)] π(x)^2 dx
Now you can evaluate this integral using the limits [(−3/11),(1/22)]. The resulting value will give you the volume of the solid generated by the region bounded by the curves.