8.4 Areas Between Curves and Lines

基本概念

学习目标：掌握曲线与直线之间面积的计算方法，结合积分与几何图形面积。

面积 = 几何图形面积 - \(\int_a^b f(x) dx\)

重要：曲线与直线围成的区域面积可以通过积分与几何图形面积相结合的方法计算。

The diagram shows a sketch of part of the curve with equation \(y = x(4 - x)\) and the line with equation \(y = x\).

Find the area of the region bounded by the curve and the line.

Area beneath curve = \(\int_0^3 (4x - x^2) dx\)

Shaded area = area beneath curve - area beneath line

Area beneath line = \(\int_0^3 x dx = \left[\frac{x^2}{2}\right]_0^3 = \frac{9}{2}\)

Area beneath curve = \(\left[2x^2 - \frac{x^3}{3}\right]_0^3 = 9\)

Shaded area = \(9 - \frac{9}{2} = \frac{9}{2}\)

The diagram shows a sketch of the curve with equation \(y = x(x - 3)\) and the line with equation \(y = 2x\).

Find the area of the shaded region OAC.

The curve cuts the x-axis at \(x = 3\) (and \(x = 0\)) so \(a = 3\).

The point C is (5, 10).

Area of triangle OBC = \(\frac{1}{2} \times 5 \times 10 = 25\)

Area between curve, x-axis and the line \(x = 5\) is:

\(\int_3^5 x(x - 3) dx = \int_3^5 (x^2 - 3x) dx\)

\(= \left[\frac{x^3}{3} - \frac{3x^2}{2}\right]_3^5\)

\(= \left(\frac{125}{3} - \frac{75}{2}\right) - \left(\frac{27}{3} - \frac{27}{2}\right)\)

\(= \frac{26}{3}\)

Shaded region = \(25 - \frac{26}{3} = \frac{49}{3}\)

Problem-solving：Look for ways of combining triangles, trapeziums and direct integrals to find the missing area.

三角形面积 = \(\frac{1}{2} \times 底 \times 高\)

梯形面积 = \(\frac{1}{2} \times (上底 + 下底) \times 高\)

矩形面积 = 长 × 宽

常见错误：

学习建议：重点掌握交点求法，多做复合图形面积计算，养成先画图的习惯。