Question

Use the method of cylindrical shells to find the volume V of the solid obtained by rotating the region bounded by the given curves about the x-axis.

x + y = 7, x = 16 − (y − 3)^2

Answer 1

The two curves meet at :
6 - y = 15 - (y - 3)²
y² - 7y = 0
y = 0 and y = 7
By the shells method ,
the radius is y ,
the height is 15 - (y - 3)² - (6 - y),
the thickness is dy ;

V = 2π ∫(y = 0 to 7) y ( 15 - (y - 3)² - (6 - y) ) dy =
= 2π ∫(y = 0 to 7) 7y² - y³ dy =
= 2π ( 7/3 y³ - 1/4 y⁴ ) | (y = 0 to 7) =
= 2π ( 7/3 × 343 - 2401/4 ) = 2401π/6

Answer 2

thanks.)