8.5 Areas Between Two Curves

基本概念

学习目标：掌握两条曲线之间面积的计算方法，使用积分公式求解。

面积 = \(\int_a^b [f(x) - g(x)] dx\)
= \(\int_a^b f(x) dx - \int_a^b g(x) dx\)

重要：只有当两条曲线在积分区间内不相交时，才能使用这个公式。

图1：两条曲线间面积的基本概念示意图

The diagram shows a sketch of part of the curve with equation \(y = 5 + 4x - x^2\) and part of the curve with equation \(y = \frac{x^2}{2} + 1\).

Find the area of the region R bounded by the two curves.

图2：Example 1 - 两条曲线围成的区域R

确定积分限：从图中可以看出积分限为1到3
应用公式：Area of R = \(\int_1^3 (5 + 4x - x^2) dx - \int_1^3 (\frac{x^2}{2} + 1) dx\)
计算第一个积分：\(\int_1^3 (5 + 4x - x^2) dx = \left[5x + 2x^2 - \frac{x^3}{3}\right]_1^3\)
计算第二个积分：\(\int_1^3 (\frac{x^2}{2} + 1) dx = \left[\frac{x^3}{6} + x\right]_1^3\)
代入计算：Area of R = 11 (units²)

The diagram below shows a sketch of part of the curve S with equation \(y = 8 - 2x - x^2\)

The curve T with equation \(y = x^2 + x + 6\) intersects S at two points.

a) Find the x-coordinates of the points of intersection of S and T.

b) Use calculus to find the area of the finite region bounded by S and T.

图3：Example 2 - 曲线S和T的交点示意图

b) Area = \(\int_{-2}^{0.5} (8 - 2x - x^2) - (x^2 + x + 6) dx\)

Area = \(\int_{-2}^{0.5} (2 - 3x - 2x^2) dx\)

Area = \(\left[2x - \frac{3x^2}{2} - \frac{2x^3}{3}\right]_{-2}^{0.5}\)

Area = \(\frac{125}{24}\)

重要提示：You will need to find the limits yourself if you are required to find the area bounded by two curves.

面积 = \(\int_a^b [f(x) - g(x)] dx\)

面积 = \(\int_a^b f(x) dx - \int_a^b g(x) dx\)

面积 = \(\int_a^b (f(x) - g(x)) dx\)

常见错误：

学习建议：重点掌握交点求法，多做两条曲线间面积计算，养成先画图的习惯。