Areas in the Plane
Area Between Curves
page 390
Area Between Curves
• If f and g are continuous with 𝑓
𝑥 ≥ 𝑔(𝑥)
throughout [a,b], then the area between the
curves 𝑦 = 𝑓 𝑥 𝑎𝑛𝑑 𝑦 = 𝑔(𝑥) from a to b is the
integral of [f – g] from a to b.
𝑏
𝐴=
𝑓 𝑥 − 𝑔 𝑥 𝑑𝑥
𝑎
Example 1:
• Find the area of the region between 𝑦 = 𝑠𝑒𝑐 2𝑥 𝑎𝑛𝑑 𝑦 =
sin 𝑥 from 𝑥 = 0 𝑡𝑜 𝑥 =
𝜋
.
4
Example 2:
• Find the area of the region enclosed by the parabola 𝑦 =
2 − 𝑥 2 and the line 𝑦 = −𝑥.
Example 3:
• Find the area of the region enclosed by the graphs of
𝑦 = 2𝑐𝑜𝑠𝑥 𝑎𝑛𝑑 𝑦 = 𝑥 2 − 1.
Boundaries with Changing Functions
• If a boundary of a region is defined by more than one
function, we can partition the regions into sub-regions
that correspond to the function changes and proceed as
usual.
Example 4:
• Find the area of the region R in the first quadrant that is
bounded by 𝑦 = 𝑥 and below by the x-axis and the line
𝑦 = 𝑥 − 2.
Example 4: using geometry
• Find the area of the region in example 4 by subtracting
the area of the triangular regions from the area under the
square root curve.
Integrating with respect to y.
• Sometimes the boundaries of a region are more easily
described by functions of y than by functions of x.
Example 5:
• Find the area of the region in example 4 by integrating
with respect to y.
• (Find the area of the region R in the first quadrant that is
bounded by 𝑦 = 𝑥 and below by the x-axis and the line
𝑦 = 𝑥 − 2.)
Example 6:
• Find the area of the region enclosed by the graphs of
𝑦 = 𝑥 3 𝑎𝑛𝑑 𝑥 = 𝑦 2 − 2.