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Area

Integration

Japanese school year: Math II

What you learn

Learn to calculate areas enclosed between curves and lines by integrating the difference between upper and lower functions over the corresponding interval. This method is widely applied in geometric measurements and engineering analysis. Mastery of definite integral calculations and finding intersection points of graphs is recommended.

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Key points

This formula calculates the area SS between the curve y=f(x)y=f(x) and the xx-axis from x=ax=a to x=bx=b. Taking the absolute value ∣f(x)∣|f(x)| ensures that regions below the xx-axis are counted with a positive area.

S=∫ab∣f(x)∣ dxS = \int_{a}^{b} |f(x)| \, dx

This formula finds the area SS between two curves from x=ax=a to x=bx=b. You integrate the difference by subtracting the lower curve g(x)g(x) from the upper curve f(x)f(x).

S=∫ab{f(x)−g(x)} dx(f(x)≥g(x))S = \int_{a}^{b} \{f(x) - g(x)\} \, dx \quad (f(x) \ge g(x))

This shortcut formula allows quick evaluation of the definite integral between two intersection points, α\alpha and β\beta. Instead of expanding the expression, you simply cube the difference β−α\beta - \alpha.

∫αβ(x−α)(x−β) dx=−16(β−α)3\int_{\alpha}^{\beta} (x - \alpha)(x - \beta) \, dx = -\frac{1}{6}(\beta - \alpha)^3

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