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. We estimate 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. |