Question

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = x², y = 4x; about the y-axis

a) V = π/3
b) V = π/6
c) V = 2π/3
d) V = 5π/6

Answer 1

To find the volume of the solid obtained by rotating the region bounded by the curves y = x² and y = 4x about the y-axis, we can use the method of cylindrical shells. Option (C) is correct.

Step-by-step explanation:

To find the volume of the solid obtained by rotating the region bounded by the curves y = x² and y = 4x about the y-axis, we can use the method of cylindrical shells. First, let's find the points of intersection of the two curves.

Set y = x² equal to y = 4x:
x² = 4x
x² - 4x = 0
x(x - 4) = 0
x = 0 or x = 4

Since we're rotating about the y-axis, the region will be bounded by the curves from x = 0 to x = 4. Now, let's find the height of each cylindrical shell. The height is the difference between the y-coordinates of the curves at a given x-value. The y-coordinate of the curve y = x² is x², and the y-coordinate of the curve y = 4x is 4x. So the height is 4x - x². Finally, we integrate the volume formula for cylindrical shells:

V = 2π ∫[from 0 to 4] (x)(4x - x²) dx