Cracking The Ap Calculus ab Exam 2018

If you slice the region vertically and revolve the slice, you won’t get a washer; you’ll get a cylinder
instead. Because each slice is an infinitesimally thin rectangle, the cylinder’s “thickness” is also very,
very thin, but real nonetheless. Thus, if you find the surface area of each cylinder and add them up, you’ll
get the volume of the region.

Why work    in  the dark?   Just    as  you spend   time    practicing

formulas    in  order   to  memorize    them,   make    sure    you actually

work    on  drawing these   examples.   If  you can’t   visualize   the

problem,    you won’t   be  able    to  set up  the integral.

The formula for the surface area of a cylinder is 2πrh. The height of the cylinder is the length of the
vertical slice, (2 − x^2 ) − x^2 = 2 − 2x^2 , and the radius of the slice is x. Thus, evaluate the integral.

2 π x(2 −   2x^2 )  dx

The math goes like the following:

2 π x(2 −   2x^2 )  dx  =   2π (2x  −   2x^3 )  dx  =   2π  =   π

Suppose you tried to slice the region horizontally and use washers. You’d have to convert each equation
and find the new limits of integration. Because the region is not bounded by the same pair of curves
throughout, you would have to evaluate the region using several integrals. The cylindrical shells method
was invented precisely so you can avoid this.