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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.