8.4 Areas Between Curves and Lines

Exercise 8D - 曲线与直线间面积

The diagram shows part of the curve with equation \(y = x^2 + 2\) and the line with equation \(y = 6\).

The line cuts the curve at the points A and B.

a) Find the coordinates of the points A and B.

b) Find the area of the finite region bounded by line AB and the curve.

提示：先求交点，然后计算矩形面积减去曲线下面积。

答案：
a) A(-2, 6), B(2, 6)
b) Area = 16 - \(\frac{32}{3}\) = \(\frac{16}{3}\)

The diagram shows the finite region, R, bounded by the curve with equation \(y = 4x - x^2\) and the line \(y = 3\).

The line cuts the curve at the points A and B.

a) Find the coordinates of the points A and B.

b) Find the area of R.

提示：先求交点，注意曲线开口向下，需要计算矩形面积减去曲线下面积。

答案：
a) A(1, 3), B(3, 3)
b) Area = 6 - \(\frac{4}{3}\) = \(\frac{14}{3}\)

The diagram shows a sketch of part of the curve with equation \(y = 9 - 3x - 5x^2 - x^3\) and the line with equation \(y = 4 - 4x\).

The line cuts the curve at the points A and B.

a) Find the coordinates of the points A and B.

b) Find the area of the finite region bounded by the curve and the line.

提示：先求交点，然后计算三角形面积减去曲线下面积。

答案：
a) A(-1, 8), B(1, 0)
b) Area = 4 - \(\frac{8}{3}\) = \(\frac{4}{3}\)