Find the volume of the solid formed by rotating the region bounded by the graph of y equals 1 plus the square root of x, the y-axis, and the line y = 3 about the line y = 5.
draw the region. Using discs (washers), the volume is
v = ∫[0,4] π(R^2-r^2) dx
where R = 5-y and r = 5-3=2
v = ∫[0,4] π((5-(1+√x))^2-(2)^2) dx = 40π/3
using cylinders, we have
v = ∫[1,3] 2πrh dy
where r=5-y and h=x=(y-1)^2
∫[1,3] 2π(5-y)(y-1)^2 dy = 40π/3
Thank you steve.
To find the volume of the solid formed by rotating the region bounded by the graph of y = 1 + √x, the y-axis, and the line y = 3 about the line y = 5, we can use the method of cylindrical shells.
1. First, sketch the given region on a graph to visualize it.
2. The region is bounded by the graph of y = 1 + √x, the y-axis, and the line y = 3. The region starts at x = 0 and ends at the x-coordinate where 1 + √x = 3. So, we need to solve the equation 1 + √x = 3 for x to find the limits of integration.
Subtracting 1 from both sides, we get √x = 2.
Squaring both sides, we get x = 4.
Hence, the limits of integration for x are 0 to 4.
3. Now, consider a vertical strip at an x-distance x from the y-axis. This strip will have a width dx and a height of y = 1 + √x.
The distance between the line y = 5 and the strip is (5 - y). This distance represents the radius of the cylindrical shell.
Therefore, the volume of the cylindrical shell is given by 2π(5 - y)dx.
4. To find the total volume, we integrate the expression for the volume of each cylindrical shell over the range of x from 0 to 4.
Volume = ∫[0 to 4] 2π(5 - y)dx
5. Substitute y = 1 + √x into the integral:
Volume = ∫[0 to 4] 2π(5 - (1 + √x))dx
6. Simplify the expression:
Volume = ∫[0 to 4] 2π(4 - √x)dx
7. Integrate the expression:
Volume = 2π ∫[0 to 4] (4 - √x)dx
Volume = 2π [4x - (2/3)x^(3/2)] from x = 0 to x = 4
8. Evaluate the integral:
Volume = 2π [(4(4) - (2/3)(4)^(3/2)) - (4(0) - (2/3)(0)^(3/2))]
Volume = 2π [(16 - 32/3) - (0 - 0)]
Volume = 2π [(48/3 - 32/3)]
Volume = 2π [16/3]
Volume = (32π/3)
Hence, the volume of the solid formed by rotating the region is (32π/3) cubic units.