a in terms of V
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 the line x=a from b to c is
V= ∫bc2π a-x f(x)dx
The triangle with the vertices (0, 0), (1, 0), (1, 2) rotated about the line x=a
The triangle with the vertices (0, 0), (1, 0), (1, 2) is shown in figure.
First, join the given points.
From the figure, the boundary curves are y=2x, y=0 and x=2
Therefore, the region of the triangle T is bounded by y=2x and x=2 rotated about the line x=a.
Using the shell method, find the typical approximating shell is with radius (a-x),
Therefore, the circumference is 2π(a-x) and the height is y=2x
So, the total volume is
V= ∫ab2π(a-x) (2x)dx
Solve the equation for a in terms of V.
a in terms of V is a=V2π+23