5.3
曲線間の面積：問題解決
領域は、平方根関数、反転した立方根関数、そして線形関数の3つの曲線で囲まれます。線形関数は他の2つの曲線とそれぞれ交差し、これらの交点によって囲まれた領域の境界が変化する位置が決まります。グラフの異なる部分では、異なる曲線が上限と下限の境界となるため、区間全体にわたって単一の積分設定で面積を求めるこ…
平方根関数、反射立方根関数、線形関数で囲まれた領域を考えます。
線形関数は他の2つの曲線と交わり、領域の境界を形成します。
面積を求めるために、交点での境界曲線の変化に基づいて領域を2つのセグメントに分けます。
x = 0 から x = 1 までの区間では、xに対する上限と下限の境界曲線間の垂直距離を積分し、この区間上の定積分を評価することで面積を求めます。
x = 1からx = 4までの区間では、異なる境界曲線のペアが領域の上と下を定義します。この区間の面積は、対応する垂直距離を積分し、定積分を算ることでも求められます。
総囲い面積は2つの定積分の和です。
この方法は福祉経済学で総経済余剰を測定するために用いられます。需要曲線と供給曲線が地域の境界を形成しています。総余剰は、0から均衡量までの曲線間の垂直差を積分することで計算され、これが純経済的利益を表します。
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Q1: Why do you need to split a region into multiple segments when finding the area between curves?
When different curves form the upper and lower boundaries in different parts of a region, a single integral cannot accurately represent the entire area. Splitting at intersection points ensures each segment maintains a consistent upper-minus-lower relationship, which is necessary for accurate geometric interpretation and correct calculation of the total enclosed area.
Q2: How do you calculate the area in each segment of a piecewise region?
For each segment, integrate the vertical distance between the appropriate upper and lower boundary curves with respect to x, then evaluate the definite integral over that interval. The total area is found by adding the results from all segments. This method ensures each integral represents a single, consistent relationship between the curves.
Q3: What role do intersection points play in setting up the integrals for area problems?
Intersection points determine where the boundary curves change roles as upper and lower limits. These points mark the boundaries between segments and define the intervals over which each integral is evaluated. Identifying intersection points is essential for correctly dividing the region and setting up accurate integral expressions.
Q4: How is the piecewise area method applied in welfare economics?
In welfare economics, demand and supply curves form the boundaries of the surplus region, with the equilibrium point determining the integration interval. Total economic surplus is calculated by integrating the vertical difference between demand and supply from zero to equilibrium quantity, representing the net economic benefit generated by market exchange.
Q5: What does the area between demand and supply curves represent economically?
The area between demand and supply curves represents total economic surplus, combining consumer and producer gains into a single measure. This area quantifies the net economic benefit generated by market exchange at the equilibrium point, showing the total welfare gain from trade.
Q6: How do you identify which curve is the upper boundary in each segment?
Examine the curves in each interval between intersection points. The curve that lies above the other in that specific interval is the upper boundary. This may change from one segment to the next, which is why splitting the region at intersection points is necessary for correct setup.
Q7: Why is it important to evaluate definite integrals separately for each segment?
Evaluating definite integrals separately for each segment ensures that each integral uses the correct upper and lower boundary curves for its interval. This prevents sign errors and geometric misinterpretation. Summing the separate results gives the accurate total enclosed area of the region.