Graph the integrand and use known area formulas to evaluate the integrals.

Question

Graph the integrands and use known area formulas to evaluate the integrals

∫-1 to 1 (2-abs(x))dx

Answer 1

The graph of y = 2 - l x l has highest point (0,2) and has the following shape: ∧

Between x = -1 and x = 1, the graph lies above the x-axis.

So, ∫_{(-1 to 1)} [ 2 - l x l ]dx = area bounded above by the graph of y = 2 - l x l and below by the x-axis.

= [Area of triangle with vertices (0,2), (-1,1), and (1,1)] + [Area of rectangle with

vertices (-1,0), (-1,1), (1,1), and (1,0)]

= 3