Let f (x) = x3. Use a left-hand sum with 3 sub-intervals to estimate . Is your answer an over- or under-estimate?
We use exactly the same process as we did for the non-negative functions, except now some of our "heights" will be negative.
First, split up the interval [-3,0] into 3 sub-intervals of size 1.
On each sub-interval, find the value of f at the left-endpoint of the sub-interval and use it to draw a rectangle.
Multiply each value of f by the length of its sub-interval, and add the results:
f (-3)(1) + f (-2)(1) + f (-1)(1) = (-27)(1) + (-8)(1) + (-1)(1) = -36.
Or, do the shortcut: add the values of f, then multiply by the length of a sub-interval:
[f (-3) + f (-2) + f (-1)](1) = [-27-8-1](1) = -36.
The left-hand sum rectangles cover more area than we would like. However, since this area is all below the x-axis, the left-hand sum gives us a more negative value than the actual integral. This means the left-hand sum is an underestimate.