Calculus 2: Volumes and Volumes by Cyl. Shells HW problem.

Question

Problem:

Graph the following enclosed region:

y=-x^2+11x , y=0, x=0

State the method that be used to find the volume of the solid created by rotating the region about the following axis of rotation:

a) y-axis(line x=0) b) line y=2

For each part above draw the region with a representation rectangle and label the endpoints of the interval spanned and any radius and/or height needed in finding the volume

a) b)

Answer 1

-x² + 11x = 0; x = 0 or x = 11

a) V = 2π∫₀¹¹x(- x² + 11x)dx = 2π(- x⁴/4 + 11x³/3)₀¹¹ = 2π·11⁴(-1/4 + 1/3) = 14641π/6

b) Line y = 2 intercepts region and because somthiing wrong