Barrons AP Calculus

Figure  N7–4

A2. Using Symmetry

Frequently we seek the area of a region that is symmetric to the x- or y-axis (or
both) or to the origin. In such cases it is almost always simpler to make use of
this symmetry when integrating. For example: ■ The area bounded by the x-axis
and this arch of the cosine curve is symmetric to the y-axis; hence it is twice the
area of the region to the right of the y-axis.

■   The area    bounded by  the parabola    and the line    is  symmetric   to  the x-axis;

hence   it  is  twice   the area    of  the region  above   the x-axis.