The volume of solid using the midpoint rule.
i. If x is the radius of the typical shell, then the circumference =2πx and the height is y=f(x)
ii. By the shell method, the volume of the solid by rotating the region under the curve y=f(x) about x-axis from a to b is
V= ∫ab2πx f(x)dx
iii. To approximate the integral is same as finding the Riemann sum for the midpoints. Use the formula and find the value of the integral.
The Riemann sum for the midpoints:
where, xi-=12xi-1+xi and ∆x=b-an, n is the number of the subintervals.
The volume of the solid is
Assume that, gx=2πx fx
So, volume of solid is
where x is the radius and fx is the height of each cylinder.
where yi is the upper part of the region and yi-1 is the lower part of the region
From the given figure,
The region is bounded between 0 to 10
That means a=0, b=10 and n=5
So the subintervals are 0, 2, 2, 4, 4, 6, 6, 8, 8, 10
Then the midpoints will be 1, 3, 5, 7, 9
By using the midpoint rule, the volume of the solid is
Substitute gx=2πx fx,
To write the values of fxi from the given figure,
The volume of the solid obtained by rotating the given region about the y-axis is