Suppose the area under y = -x^2+1 between x = 0 and x = 1 is rotated around the x-axis. Find the volume by using Disk method and shell method.

discs of thickness dx:

v = ∫[0,1] πr^2 dx
where r=y = 1-x^2
v = ∫[0,1] π(1-x^2)^2 dx

shells of thickness dy:
v = ∫[0,1] 2πrh dy
where r=y and h=x=√(1-y)
∫[0,1] 2πy√(1-y) dy
To integrate, use
u^2 = 1-y
dy = -2u du

To find the volume by using the Disk method, we need to integrate the area of the disks formed by the rotation.

Step 1: Identify the equation and limits of integration.
The given equation is y = -x^2 + 1, and the limits of integration are x = 0 and x = 1.

Step 2: Determine the radius and height of the disks.
For the Disk method, we consider the radius and the height of the disks.
- The radius of each disk is the distance from the axis of rotation (x-axis) to the curve y = -x^2 + 1.
- The height of each disk is the thickness of the disk, which is dx.

Step 3: Find the radius of each disk using the given equation.
The radius of each disk is the y-value of the curve y = -x^2 + 1 at a specific x-value.
For this problem, the radius, r, is given by r = -x^2 + 1.

Step 4: Write the formula for the volume of each disk.
The volume of each disk, dV, is given by dV = π * (radius)^2 * dx.
Substituting the radius, we have dV = π * (-x^2 + 1)^2 * dx.

Step 5: Integrate to find the total volume.
To find the total volume, we integrate the formula for the volume of each disk. The integral will be taken with respect to x, from x = 0 to x = 1.
V = ∫[0,1] π * (-x^2 + 1)^2 * dx.

Now, let's find the volume using the Shell method.

Step 1: Determine the height and radius of the shells.
For the Shell method, we consider the height and radius of the shells.
- The height of each shell is the difference between the two curves: y = -x^2 + 1 and y = 0.
- The radius of each shell is the distance from the axis of rotation (x-axis) to the curve y = -x^2 + 1.

Step 2: Determine the height of each shell using the given equation.
The height of each shell is given by h = (-x^2 + 1) - 0 = -x^2 + 1.

Step 3: Determine the radius of each shell.
The radius of each shell is the x-value at a specific height. In this problem, the height is given by h = -x^2 + 1. Solving for x, we get x = ±sqrt(1 - h).

Step 4: Write the formula for the volume of each shell.
The volume of each shell, dV, is given by dV = 2π * radius * height * dx.
Substituting the radius and height, we have dV = 2π * (-x^2 + 1) * (sqrt(1 - h)) * dx.

Step 5: Integrate to find the total volume.
To find the total volume, we integrate the formula for the volume of each shell. The integral will be taken with respect to h, from h = 0 to h = 1.
V = ∫[0,1] 2π * (-x^2 + 1) * (sqrt(1 - h)) * dx.

Note that for the shell method, we integrate with respect to the "height" variable, which is h in this case, instead of x.

To find the volume of a solid formed by rotating a curve about an axis, we can use the disk method or the shell method.

1. Disk Method:
The disk method involves dividing the shape into infinitesimally thin disks perpendicular to the axis of rotation. The volume of each disk is a small cylinder whose height is given by the function and whose radius is the distance from the curve to the axis of rotation.

For the given function y = -x^2 + 1 and the interval x = 0 to x = 1, we need to find the volume when rotated around the x-axis.

Step 1: Determine the interval of integration.
In this case, the interval is given as x = 0 to x = 1.

Step 2: Set up the integral for a single disk.
The volume of a single disk is given by V = π * r^2 * h, where r is the radius of the disk and h is the height.

The radius of the disk is the distance from the curve to the x-axis, which is y. Therefore, r = y = -x^2 + 1.

The height of the disk is an infinitesimally small change in x, which is dx.

Step 3: Set up the integral for the entire solid.
Integrate the volume of each disk over the interval x = 0 to x = 1:
V = ∫[0, 1] π * (-x^2 + 1)^2 * dx

Evaluate the integral to find the volume.

2. Shell Method:
The shell method involves dividing the shape into infinitesimally thin cylindrical shells parallel to the axis of rotation. The volume of each shell is a cylindrical strip whose height is the function and whose radius is the distance from the curve to the axis of rotation.

For the given function y = -x^2 + 1 and the interval x = 0 to x = 1, we need to find the volume when rotated around the x-axis.

Step 1: Determine the interval of integration.
In this case, the interval is given as x = 0 to x = 1.

Step 2: Set up the integral for a single shell.
The volume of a single shell is given by V = 2π * r * h * w, where r is the radius of the shell, h is the height, and w is the width or thickness.

The radius of the shell is the distance from the curve to the x-axis, which is x. Therefore, r = x.

The height of the shell is the function, y = -x^2 + 1.

The width or thickness of the shell is an infinitesimally small change in x, which is dx.

Step 3: Set up the integral for the entire solid.
Integrate the volume of each shell over the interval x = 0 to x = 1:
V = ∫[0, 1] 2π * x * (-x^2 + 1) * dx

Evaluate the integral to find the volume.

By finding the volume using both the disk method and the shell method, we can verify our results and ensure accuracy.