Calculus; Need 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. 

Answer Choices:

	I only
	II only
	III only
	I and III

Answer 1

Last integral is right gives volume of the solid gives by Disk Method integral I

Answer 2

Hi Lilah. This is a standard use of the formula for a volume of revolution,

Rather, than just give the answer, let's do it the long way.

First, draw a diagram. You will see that the area to be rotated around the X-axis is a "triangle" with sides the vertical line x = 03 the X-axis, and the parabola y = x², from x = 0 to x = 3.

Imagine a thin slice cut out of this area at the horizontal position x = x, up to the parabola, and of thickness dx.

Now, when we rotate this slice around the X-axis, it forms a thin disk of radius, y, and thickness, dx.

The volume of this disk is dV = π y² dx

So, the entire volume = V = ∫dV = ∫ {from x = 0 to 3} π y² dx

= π ∫ {from x = 0 to 3} π x⁴ dx . . . . . . . . y = x² => y² = x⁴

So, only option 1 is correct.