f(x) dx
These integrals can be evaluated by finding the area under the curve (and above the x-axis) on the intervals. The areas can be calculated by breaking them up into familiar shapes.
a) The area is a trapezoid with vertical line segment bases and a horizontal height. The bases are 5 units and 15 units long, and the height of the trapezoid is 10 units. So the integral = [(5+15)/2]*10 = 100 square units.
b) Since we need from 0 to 25, the easiest way may be to add the answer from part A to the area of the trapezoid that represents the integral from 10 to 25. This second trapezoid has horizontal bases of 5 and 15 units and a vertical height of 15, so the integral = 100 + [(5+15)/2*15] = 250 square units.
c) This time we are looking for an area that is entirely beneath the x-axis, so the answer will be negative. The shape is a triangle with base 10 and height -15, so the integral = (10*-15)/2 = -75 square units.
Hope this helps!
