Answer:
the correct answer is -π/2
Step-by-step explanation:
To find the volume of the solid of revolution formed by revolving the region bounded by the curve y=x² and the x-axis between x=0 and x=1 around the line y=1, we can use the method of cylindrical shells.
The first step is to find the height of each shell. Since we are revolving the region around the line y=1, the height of each shell will be the difference between the y-value of the curve y=x² and the line y=1. So, the height of each shell is (x² - 1).
Next, we need to find the radius of each shell. The radius is simply the x-value of the curve y=x². So, the radius of each shell is x.
Now, we can set up the integral to calculate the volume of the solid of revolution. The integral is:
V = 2π ∫[0 to 1] (x)(x² - 1) dx
Evaluating this integral, we get:
V = 2π ∫[0 to 1] (x³ - x) dx
= 2π [ 1/4 x^4 - 1/2 x^2 ] [0 to 1]
= 2π [ (1/4 - 1/2) - (0 - 0) ]
= 2π [ -1/4 ]
= -π/2
The volume of the solid of revolution is -π/2.
Therefore, the correct answer is -π/2
