Calculus help please

Question

. 

Which of the following statements is true? The volume of the solid formed by rotating the region bounded by the graph of y = x², x = 3, y = 0 around the x-axis is (2 points)

1. 
2. 
3. 

	I only
	II only
	III only
	I and III

Answer 1

The answer is choice 1. The definition of calculating the volume generated by a function by revolution about a given axis is given by:

1. About the x-axis:
2. Volume = π · ∫ [ f(x) ]² dx
3. About the y-axis:
4. Volume = π · ∫ [ f(y) ]² dy

For the given problem:

1. f(x) = y = x²
2. x = 3
3. y = 0

Therefore, the integral is evaluated from 0 to 3. This is found by setting f(x) to y to find the boundaries:

1. y = f(x)
2. 0 = x²
3. x = 0

Knowing these boundaries, and plugging the values which have been given, one finds that the equation to compute the volume by rotation is given by:

Volume = π · ∫ [ f(x) ]² dx = π · ∫ [ x² ]² dx = π · ∫ x⁴ dx

I am unable to write the limits of integration but they need to go from 0 to 3, inclusive... [0, 3]