for the region bounded by y=x^3, y=0,x=0,x=3 find the following:

Question

a) area by integrating relative to x and y

b) volume by rotating around the x axis using 1) disc or washers, 2)shells

c) volume by rotating around y axis integrating relative to x and y

d) volume around rotating around line x=5 using any method

e) volume by rotating around the line x=27 using any method

f) the arc length of y=x^3 from x=0 to 3

g) the area of surface of revolution by rotating around x axis

 

I thank you in advance for pointing me in the right direction

Answer 1

a)  Area:

∫₀³ x³dx=x⁴/4|₀³=3⁴/4=81/4;

b)If you revolve around x-axis the formula to find Volume is:

V_disk=π∫₀³[f(x)]²dx=π∫₀³x⁶dx=(π/7)3⁷=..calculate

V_shell=π∫₀³xf(x)dx=π∫₀³x⁴dx=(π/5)3⁵=..calculate

c)Rotating arouud y axis is similar

i) y-variable; x=y^(1/3)

The rest of the question is, to some extent, similar, So try.