The volume obtained by rotating about the y-axis the region under the given curve.
i. If x is the radius of the typical shell, then the circumference =2πx and the height is y.
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
iii. To approximate the integral is the 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. And xi=a+i∆x
As the region is bounded by
Draw the region using the given curves.
The graph shows the region and the height of cylindrical shell formed at x by the rotation about the line y-axis
It has the radius x.
Therefore, the circumference is 2πx and the height is 1+x3
It is given that 0≤x≤1
So, a=0 and b=1
So, the volume of given solid is
For the midpoint rule,
Find ∆x &xi-.
0≤x≤1, means a=0, b=1 and n=5
So, the subintervals are 0, 0.2, 0.2, 0.4, 0.4, 0.6, 0.6, 0.8, 0.8, 1
Then the midpoints will be 0.1, 0.3, 0.5, 0.7, 0.9
By using the midpoint rule, the volume is
The volume of the solid obtained by rotating the region under the given curves about the y-axis is