Barrons AP Calculus - David Bock

See Questions 4, 6, 7, and 9 in the Practice Exercises.

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.


• The ellipse is symmetric to both axes; hence the area inside the ellipse is four times the area in the
  first quadrant.