Evaluate the integral by interpreting it in terms of areas. Sketch the region.

Question

6

∫ |x-4|dx

0

Answer 1

The graph of the function y = l x - 4 l is "V shaped" with lowest point at (4,0),

The value of the given integral is the area of the region below the graph and above the x-axis between x = 0 and x = 6.

The region consists of two right triangles. The first has horizontal leg with endpoints (0,0) and (4,0) and vertical leg with endpoints (0,0) and (0, 4). The other triangle has horizontal leg with endpoints (4,0) and (6,0) and vertical leg with endpoints (6,0) and (6,2).

∫_{(from 0 to 6)} l x - 4 l dx = (1/2)(4)(4) + (1/2)(2)(2) = 10