IMathAS-libretexts

The region between the lines

\displaystyle {x}={a}

and

\displaystyle {x}={b}

, and the graphs of

\displaystyle {y}={f{{\left({x}\right)}}}

and

\displaystyle {y}={g{{\left({x}\right)}}}

is rotated around the x-axis to form a solid, as shown in the figure.

Which of the following integrals represents the volume of this solid?

$\displaystyle {\int_{{a}}^{{b}}}{2}\pi{x}{\left[{f{{\left({x}\right)}}}-{g{{\left({x}\right)}}}\right]}\ {\left.{d}{x}\right.}$
$\displaystyle {\int_{{a}}^{{b}}}\pi{\left({f{{\left({x}\right)}}}-{g{{\left({x}\right)}}}\right)}^{{2}}\ {\left.{d}{x}\right.}$
$\displaystyle {\int_{{a}}^{{b}}}\pi{\left[{\left({f{{\left({x}\right)}}}\right)}^{{2}}-{\left({g{{\left({x}\right)}}}\right)}^{{2}}\right]}\ {\left.{d}{x}\right.}$
$\displaystyle {\int_{{a}}^{{b}}}{\left[{\left({f{{\left({x}\right)}}}\right)}^{{2}}-{\left({g{{\left({x}\right)}}}\right)}^{{2}}\right]}\ {\left.{d}{x}\right.}$