Sammy has no reason to be troubled by this discrepancy. He should give himself credit for calculating the definite integral correctly, which does in fact = 0. He should re-examine his geometrical analysis, though: the line y = x-3 forms two triangles with the x-axis, the y-axis (ie x = 0), and the vertical line x = 6. Each triangle has height = 3 and base = 3 and area = 9/2. Summing these two areas gives A = 9.
The fact that this area is not equal to the definite integral is because some of this area lies below the x-axis. Definite integrals register any area below the x-axis as negative, since the y-values of the function which are the heights of the Riemann rectangles are negative. Since there is exactly the same amount of "negative" area as positive area, they are cancelling to give 0 for the definite integral.
To make these two analyses agree, Sammy can use a calculator to integrate the absolute value of the function instead: ∫06 |x - 3|dx = 9 = sum of the areas of the two triangles.
To understand one simple application of this difference, he should consider a rectilinear motion example in which an object moves along a horizontal desktop with its velocity at time t seconds given by v(t) = t - 3. He might note that the object's velocity is negative for 1/2 the time, which means the object is moving left, and positive for the other 1/2, when the object is moving right.
In this instance, the integral of v(t) on [0,6] = 0, which gives us the accurate calculation of the object's displacement (it is in the same position at t = 0 secs and t = 6 secs). However, the integral of its speed, |v(t)| , on [0,6] gives us total distance traveled, which is 9.
