第5章積分

Evaluating definite integrals this way can be quite tedious because of the complexity of the calculations. Later in this chapter we develop techniques for evaluating definite integrals without taking limits of Riemann sums. However, for now, we can rely on the fact that definite integrals represent the area under the curve, and we can evaluate definite integrals by using geometric formulas to calculate that area. We do this to confirm that definite integrals do, indeed, represent areas, so we can then discuss what to do in the case of a curve of a function dropping below the x-axis.

When we defined the definite integral, we lifted the requirement that $f\left(x\right)$ be nonnegative. But how do we interpret “the area under the curve” when $f\left(x\right)$ is negative?

Let us return to the Riemann sum. Consider, for example, the function $f\left(x\right)=2-2x^2$ (shown in Figure 5.17) on the interval $\left[0,2\right].$ Use $n=8$ and choose $\{x_i^{*}\text{}}$ as the left endpoint of each interval. Construct a rectangle on each subinterval of height $f\left(x_i^{*}\right)$ and width Δx. When $f\left(x_i^{*}\right)$ is positive, the product $f\left(x_i^{*}\right)\text{Δ}x$ represents the area of the rectangle, as before. When $f\left(x_i^{*}\right)$ is negative, however, the product $f\left(x_i^{*}\right)\text{Δ}x$ represents the negative of the area of the rectangle. The Riemann sum then becomes

Taking the limit as $n\to \infty ,$ the Riemann sum approaches the area between the curve above the x-axis and the x-axis, less the area between the curve below the x-axis and the x-axis, as shown in Figure 5.18. Then,

The quantity $A_1-A_2$ is called the net signed area.

Notice that net signed area can be positive, negative, or zero. If the area above the x-axis is larger, the net signed area is positive. If the area below the x-axis is larger, the net signed area is negative. If the areas above and below the x-axis are equal, the net signed area is zero.