Question

2.) Use any method to find the volume of the solid rotated about the line x=2, whose area is bounded by y=x^(1/3) and y=x^2

Answer 1

we know that according to shell method volume of the solid generated by revolving the region bounded by y =f(x) and y = g(x) between x = a and x = b about line x = c is given by,

    -----------------------------------------------------------------1)

where,

f(x) is the top curve above g(x) between x = a and x = b and we have a < b c

we have,

Hence we can say that,

Hence we can say that x ranges from x = 0 and x = 1 and as x ranges from x = 0 to x = 1 we have a = 0 and b = 1

we can see that y = x^1/3 is the top curve above y = x² between x =0 and x = 1 hence we have f(x) = x^1/3 and g(x) = x²

solid is rotated about x = 2 hence we have c = 2

0 < 1 < 2 means a < b < c

Hence using formula 1) we can say that volume is given by,