Application of Integrals

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Area = ∫[a to b] f(x) dx, where the curve is above the x-axis. This represents the sum of infinitely many thin rectangular strips of width dx and height y = f(x).

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Area = ∫[c to d] g(y) dy. Here we use horizontal strips instead of vertical strips, integrating with respect to y from y = c to y = d.

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An elementary area strip is a thin vertical rectangle of height y and width dx. Its area is dA = y dx = f(x) dx. The total area is found by summing all such strips using integration.

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When f(x) < 0, the integral ∫[a to b] f(x) dx gives a negative value. Since area is always positive, we take the absolute value: Area = |∫[a to b] f(x) dx|.

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Split the region at points where the curve crosses the x-axis. Calculate area for each part separately and add their absolute values: Total Area = |A₁| + |A₂| + ... where each Aᵢ represents area of on…

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Area = ∫[1 to 3] x² dx = [x³/3]₁³ = (27/3) - (1/3) = 9 - 1/3 = 26/3 square units.

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Area = ∫[0 to 4] √x dx = ∫[0 to 4] x^(1/2) dx = [2x^(3/2)/3]₀⁴ = (2/3)(4)^(3/2) = (2/3)(8) = 16/3 square units.

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If f(x) ≥ g(x) in the interval [a,b], then Area = ∫[a to b] [f(x) - g(x)] dx. Always subtract the lower curve from the upper curve.