Find the volume of the solid generated by revolving the region bounded by y = x 2 and the x‐axis on [0,1] about the x‐axis.

Question

i need an answer please . for a final exam

Answer 1

Split the interval [0,1] into thin slices of width dx and height y. When rotated around the x-axis these slices form discs of radius y and volume πy²dx, so the sum of these is ∫πy²dx=π∫x⁴dx=π(x⁵/5)[0,1]=π/5 (about 0.6283) cubic units, because y=x², so y²=x⁴.