5.3
곡선 사이의 넓이: 문제 풀이
어떤 영역은 제곱근 함수, 반사된 세제곱근함수, 그리고 선형 함수라는 세 개의 곡선으로 둘러싸여 있습니다. 선형 함수는 나머지 두 곡선과 각각 교차하며, 이러한 교점은 폐곡선으로 둘러싸인 영역의 경계가 변하는 지점을 결정합니다. 그래프의 서로 다른 구간에서 위쪽 경계…
제곱근 함수, 반사된 입근근, 선형 함수로 경계가 설정된 영역을 생각해 봅시다.
선형 함수는 나머지 두 곡선과 교차하여 영역의 경계를 형성합니다.
면적을 찾기 위해 경계 곡선의 변화에 따라 지역을 두 구간으로 나눕니다.
x = 0에서 x = 1까지의 구간에서, 면적은 x에 대해 상하 경계 곡선 사이의 수직 거리를 적분하고, 이 구간에서의 정분을 평가하여 구한다.
x = 1에서 x = 4까지의 구간에서는 서로 다른 경계 곡선 쌍이 영역의 상단과 하단을 정의합니다. 이 구간의 면적은 해당 수직 거리를 적분하고 정적분을 계산함으로써 구할 수 있다.
전체 밀폐 면적은 두 정분의 합이다.
이 방법은 복지경제학에서 총 경제 잉여를 측정하는 데 사용됩니다. 수요와 공급 곡선이 이 지역의 경계를 형성합니다. 총 잉여는 0에서 균형량까지의 곡선 간 수직 차이를 적분하여 계산되며, 이는 순경제적 이익을 나타냅니다.
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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.