如何计算单位正方形内两条平行线y=5x+0.21与y=5x+0.01之间的面积？

Alright, let's work through this problem step by step to find the area between those two parallel lines within the unit square (0 ≤ x,y ≤ 1).

Our two lines are:

• L1: y = 5x + 0.21
• L2: y = 5x + 0.01

Since their slope is 5 (very steep), plugging in x=1 gives y-values of 5.21 and 5.01—way above the square's upper y-limit of 1. So we need to find where each line crosses y=1:

• For L1: Solve 1 = 5x + 0.21 → x = (1 - 0.21)/5 = 0.158, so the intersection is (0.158, 1)
• For L2: Solve 1 = 5x + 0.01 → x = (1 - 0.01)/5 = 0.198, so the intersection is (0.198, 1)

At x=0, both lines have y-values within the square: (0, 0.21) for L1 and (0, 0.01) for L2. So inside the square, L1 is the segment from (0, 0.21) to (0.158, 1), and L2 is the segment from (0, 0.01) to (0.198, 1).

The area we want is all points in the square where 5x + 0.01 ≤ y ≤ min(5x + 0.21, 1). This splits into two distinct x-intervals:

• 0 ≤ x ≤ 0.158: Both lines sit below y=1 here, so the vertical distance between them is a constant 0.2 (since (5x+0.21)-(5x+0.01)=0.2).
• 0.158 < x ≤ 0.198: L1 is above y=1 (outside the square) here, so the upper boundary is y=1. The vertical distance here is 1 - (5x + 0.01) = 0.99 - 5x.

Interval 1 Area #

This is a simple rectangle—constant height over a fixed x-range:

Area₁ = (0.158 - 0) * 0.2 = 0.0316

Interval 2 Area #

We need to integrate the vertical distance over this x-range:

Area₂ = ∫₀.₁₅₈⁰.₁₉₈ (0.99 - 5x) dx

The antiderivative of 0.99 - 5x is 0.99x - 2.5x². Evaluating at the bounds:

• At x=0.198: 0.99*0.198 - 2.5*(0.198)² = 0.19602 - 0.09801 = 0.09801
• At x=0.158: 0.99*0.158 - 2.5*(0.158)² = 0.15642 - 0.06241 = 0.09401
• Subtract to get Area₂: 0.09801 - 0.09401 = 0.004

Add the two areas together to get the final result:

Total Area = 0.0316 + 0.004 = 0.0356

Quick Verification with Shoelace Formula #

We can confirm this using the shoelace formula on the polygon formed by the region's vertices: (0, 0.01), (0, 0.21), (0.158, 1), (0.198, 1), back to (0, 0.01).

Applying the formula:

Area = 1/2 * |(0*0.21 + 0*1 + 0.158*1 + 0.198*0.01) - (0.01*0 + 0.21*0.158 + 1*0.198 + 1*0)|
Area = 1/2 * |(0 + 0 + 0.158 + 0.00198) - (0 + 0.03318 + 0.198 + 0)|
Area = 1/2 * |0.15998 - 0.23118| = 1/2 * 0.0712 = 0.0356

Perfect, same result—so we know our calculation is solid.

内容的提问来源于stack exchange，提问作者user394691