Find the volume of the solid formed by rotating the region inside the first quadrant enclosed by
y = x^3
y = 16 x
about the x-axis.
2 answers
Solved
The curves intersect at (0,0) and (4,64)
Using discs of thickness dx, we have
v = ∫[0,4] π(R^2-r^2) dx
where R=16x and r=x^3
v = ∫[0,4] π((16x)^2-(x^3)^2) dx = 65536π/21
or, you can use nested shells of thickness dy, where the height of the shells is the distance between the curves.
v = ∫[0,64] 2πrh dy
where r=y and h = ∛y - y/16
v = ∫[0,64] 2πy(∛y - y/16) dy = 65536π/21
Using discs of thickness dx, we have
v = ∫[0,4] π(R^2-r^2) dx
where R=16x and r=x^3
v = ∫[0,4] π((16x)^2-(x^3)^2) dx = 65536π/21
or, you can use nested shells of thickness dy, where the height of the shells is the distance between the curves.
v = ∫[0,64] 2πrh dy
where r=y and h = ∛y - y/16
v = ∫[0,64] 2πy(∛y - y/16) dy = 65536π/21
1 answer