Find the volume:

Question

Find the volume of the wedding band formed by revolving the region bounded by the graphs of the following functions about the horizontal line y= -3. 

 

f(x)= x^2+2

and

g(x)= 4- x^2

Answer 1

The disk method seems to be more favorable for this one. A sketch shows the region is on x = [-1,1], and that x² + 2 is below 4 - x²

 

The inner radius of a vertical disk would be 3 + (x² +2) = 5 + x², and the outer radius would be 3 + (4 - x²) = 7 - x²

 

So the area of the disk (washer) itself would be Π[(7 - x²)² - (5 + x²)²]

 

Simplifying this gives Area = 24Π(1 - x²). The volume of the disk would therefore be area*thickness = (1 - x²)dx

                                                                                               ₁

Volume of wedding band = 24Π∫ ¹ (1 - x²)dx = 24Π(x - 1/3*x³ ]

                                                _{-1                                                                   -1}

 

Substituting and simplifying gives Volume = 32Π