Area Under a Curve

Business Calculus · Axiom Academy

Geometric interpretation of the definite integral One of the most important interpretations of the definite integral is as an area . Think of dividing the area into infinitely many infinitely thin rectangles. Each rectangle has width dx and height f(x). The integral "sums up" all these tiny areas. When a function goes below the x-axis, the definite integral gives signed area : Net Area (what the integral gives): Can be positive, negative, or zero Total Area (always positive): Must split integral and use absolute values In business, "below the axis" often means losses. The net integral represents profit minus loss. If you need total quantities (like total production), you need the total area, not net area. Find both the net area and total area between f(x) = x^2 - 4 and the x-axis from x = 0 to x = 3. Net area = -3 (the negative region "outweighs" the positive) Total area = 16/3 + 7/3 = 23/3 ≈ 7.67 (sum of both regions) To find the area between two curves f(x) and g(x): Where f(x) ≥ g(x) on [a, b] (f is the "top" curve) Find profit from x = 2 to x = 8. 5 Business Applications of Area Area between demand curve and market price = total benefit to consumers Area between market price and supply curve = total benefit to producers Income Inequality (Lorenz Curve) Area between perfect equality line and actual distribution = measure of inequality

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