Barrons AP Calculus

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7 Applications of Integration to

Geometry

CONCEPTS    AND SKILLS

In  this    chapter,    we  will    review  using   definite    integrals   to  find    areas   and

volumes;    specifically    area    under   a   curve,

area    between two curves,

volumes of  solids  with    known   cross   sections,

and volumes of  solids  of  revolution  (using  disks   and washers).

We’ll   also    review  related BC  topics, including

arc length;

arc  lengths,    areas,  and     volumes     involving   parametrically  defined

functions;  and area    and arc length  for polar   curves.

Also    for BC  Calculus    students,   we’ll   review  the topic   of  improper    integrals,

including

recognizing when    an  integral    is  improper

and  techniques  for     determining     whether     an  improper    integral

converges   or  diverges.

A. AREA

To  find    an  area,   we

(1)     draw    a   sketch  of  the given   region  and of  a   typical element;

(2)     write   the expression  for the area    of  a   typical rectangle;  and (3)     set up  the

definite    integral    that    is  the limit   of  the Riemann Sum of  n   areas   as  n   →   ∞.