Area Between Curves Objective To find the area

Area Between Curves

Objective To find the area of a region between two curves using integration

Area A of region bounded by the curves f(x) and g(x) and the lines x=a and x=b, where f and g are continuous and f>g, is

• Can think of this area between f and g as the area under f minus the area under g

Example 1 Find the area bounded by y = ex and y = x and between the lines x = 0 and x = 1

Example 2 Find the region bounded by the parabolas and

Example 3 Find the area bounded by the curves and

Example 4 Find the area of the region between and

Review 1) The area of the region bounded by curves f(x) and g(x), where f > g, between the point a and b, can be represented by the integral? 2) Write the integral that represents the region bounded by f(x) = x 2 - 4 x and g(x) = -x 2 + 2 x 3) Calculate the area bounded by the curves f(x) = x + 1 and g(x) = x/2 between x = 0 and x =4

With Trig Functions Find the area bounded by curves y = sin x and y = cos x between x = 0 and x = π/2

Practice Find the area between the curves y = sec 2 x and y = cos x from x = -π/4 to x = π/4

Variations Some curves are best treated as functions of y rather than x Find the area enclosed by y = x – 1 and y 2 = 2 x + 6.

Practice Find the area of the region bounded by the graphs y = x – 1 and y 2 = 3 – x

Practice Find the area bounded by the curves x = y 2 -2 and x = ey and the lines y = 1 and y = -1