Let S be a region bounded by the curve y=x+cosx
and the line y=x as shown in the given figure.
Find the volume of the solid generated when S is rotated about the x-axis.
Find the volume of the solid generated when S is rotated about the y-axis.
I assume you want just the region in QI, since otherwise the axis of rotation is inside part of the region.
So, we want the area whose vertices are (0,0), (0,1) and (π/2,π/2)
Around the x-axis, we have, using discs,
v = ∫[0,π/2] π ((x+cosx)^2 - x^2) dx
shells are not feasible, since we cannot solve for x in terms of y.
Around the y-axis, we need shells, so we can integrate along dx:
v = ∫[0,π/2] 2πx((x+cosx)-x) dx
To find the volume of the solid generated when S is rotated about the x-axis, we can use the method of cylindrical shells. Cylindrical shells are formed by rotating a vertical strip of the region about the axis of rotation.
To calculate the volume, we need to integrate the circumference of each cylindrical shell multiplied by its height (differential length of x). The radius of each shell can be determined by subtracting the x-coordinate of the curve y = x + cos(x) from the x-coordinate of the line y = x. The height of the shell is dx.
Let's go through the steps to find the volume:
Step 1: Determine the limits of integration. To find the limits, we need to solve the equation: y = x + cos(x) and y = x.
Setting these equations equal to each other: x + cos(x) = x.
Simplifying, we get: cos(x) = 0.
So, x = π/2, 3π/2.
Step 2: Calculate the radius of each cylindrical shell. The radius, denoted by 'r,' is given by r = x - (x + cos(x)) = -cos(x).
Step 3: Write the formula for the volume of each cylindrical shell. The formula is:
dV = 2πrh dx, where 'h' represents the height (dx) and 'r' is the radius.
Step 4: Integrate the formula to find the total volume. The integral is:
V = ∫(dV) = ∫(2π(-cos(x))(x)dx) from x = π/2 to x = 3π/2.
Simplifying, we have:
V = 2π ∫(-x cos(x)) dx from x = π/2 to x = 3π/2.
Now, to find the volume of the solid generated when S is rotated about the y-axis, we can use the method of disks/washers. Disks/washers are formed by rotating a horizontal strip of the region about the axis of rotation.
Repeat steps 1-3 from the previous solution, with the limits of integration being y = x + cos(x) and y = x.
Step 4: Integrate the formula to find the total volume. The integral is:
V = ∫(dV) = ∫(π(r^2)y dx) from y = 0 to y = 2π.
Simplifying, we have:
V = π ∫((x + cos(x))^2) dx from x = 0 to x = 2π.
By following these steps, you can find the volumes of the solids generated when S is rotated about the x-axis and the y-axis.