Question

How do you do this question?

Answer 1

Answer:

V = 91π / 6

Step-by-step explanation:

When we revolve the region about y = 2, we get a cylindrical shape on its side.  If we slice the shape into concentric shells, then each shell will have a radius of 2 − y, a thickness of dy, and a length of 7 − x.  The volume of the shell is:

dV = 2π r h t

dV = 2π (2 − y) (7 − x) dy

x = 7y², so:

dV = 2π (2 − y) (7 − 7y²) dy

dV = 14π (2 − y) (1 − y²) dy

dV = 14π (2 − 2y² − y + y³) dy

The total volume is the sum of all the shells from y=0 to y=1.

V = ∫ dV

V = ∫₀¹ 14π (2 − 2y² − y + y³) dy

V = 14π (2y − ⅔ y³ − ½ y² + ¼ y⁴) |₀¹

V = 14π (2 − ⅔ − ½ + ¼)

V = 91π / 6