Use calculus to find the exact shaded area between the curve y = −2e^(3x) + 4 and line L, which meet at x = −1 and x = 0, giving the answer as (a + be^(−c))/c.

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Area under S = ∫ from −1 to 0 of (−2e^(3x) + 4) dx = [−(2/3)e^(3x) + 4x] from −1 to 0 = −2/3 − (−(2/3)e⁻³ − 4) = 10/3 + (2/3)e⁻³ At x = 0, y = 2; at x = −1, y = 4 − 2e⁻³ Trapezium under L = ½ × 1 × (2 + 4 − 2e⁻³) = 3 − e⁻³ Shaded area = 10/3 + (2/3)e⁻³ − (3 − e⁻³) = (1 + 5e⁻³)/3 Answer: (1 + 5e⁻³)/3

4PM1/02R Jun 2024, Question 9: Definite Integrals and Areas

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