Find the area

Question

Find the area of the region bounded by the graphs of y = x, y = -x + 4, and y = 0.

Answer 1

Since the region bounded by the given curves is a triangle, the area can be found by basic geometry, as given in the previous answer.

 

If you are required to use Calculus, draw the region.  The lines y = x and y = 4-x intersect when x = 4-x.  So, x = 2

 

Between x = 0 and x = 2, the region is bounded above by y = x and below by y = 0.

 

Between x = 2 and x = 4, the region is bounded above by y = 4-x and below by y = 0.

 

So, Area = ∫_{(from 0 to 2)} xdx + ∫_{(from 2 to 4)}(4-x)dx

 

             = ½(x²)_{(from 0 to 2)} + (4x-½(x²))_{(from 2 to 4)}

 

             = 2 + 2 = 4 

 

 

Answer 2

Once mapped properly on an x-y plane coordinate, the area is contained into an isosceles triangle with vertices (0,0) ; (4,0) and (2,2). Area = 1/2 (basis times height) = (0.5)(4)(2) = 4 units of measurement .

 

Δ [(0,0); (2,2); (4,0)]