The graph of f is shown. Evaluate each integral by interpreting it in terms of areas.

Question

The graph of f is shown. Evaluate each integral by interpreting it in terms of areas.

(a) integral 0 to 4 f(x) dx = ?

(b) integral 0 to 10 f(x) dx = ?

(c) integral 10 to 14 f(x) dx = ?

(d) integral 0 to 18 f(x) dx = ?

4
f(x) dx
0

Answer 1

The grid spacing is 2 units apart. I believe they require the answer geometrically, not via solving a definite integral meaning find the area of geometric shapes under the curve.

a) Integral x = 0 to 4: Area of rectangle between x = [0,4] and y = [0,2] plus the area of the triangle between x = [0,4] and y = [2,6]. Area of rectangle = 4x2 = 8; Area of triangle = 0.5x4x4 = 8. Integral = 16.

b) Integral x = 0 to 10: Add answer from a) for x = 0 to 4 plus area of rectangle between x = [4,6] and y = [0,6] and triangle between x = [6,10] and y = [6,10]. Area from part a) = 16. Area of rectangle = 2x6 = 12. Area of triangle = 0.5*4*6 = 12. Integral = 16+12+12 = 40.

c) Integral x = 10 to 14: Area of triangle (noting y is negative) between x = [10,14] and y = [0,-6]. Area of triangle = Integral = 0.5*4*(-6) = -12.

d) Integral of x = 0 to 18: Sum of a), b), c) and the triangle between x = [14,18] and y = [-6,-4] (again noting y is negative in this region therefore the integral will be negative. Integral = 16 + 40 - 12 - 0.5*4*2 = 40.