Application of Definite Integration — Important Questions
Maharashtra Board · Class 12 · Mathematics & Statistics -Commerce
45 important questions from Application of Definite Integration for Maharashtra Board Class 12 Mathematics & Statistics -Commerce, with answers.
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Important Questions from Application of Definite Integration
Which of the following represent the correct setup for finding area between two curves y = x² and y = 4?
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2∫₀² (4 - x²) dx, ∫₋₂² (4 - x²) dx
The curves intersect at x = ±2. The line y = 4 is above the parabola y = x² in this interval. By symmetry, we can use 2∫₀² (4 - x²) dx or directly ∫₋₂² (4 - x²) dx.
The area of the ellipse x²/9 + y²/16 = 1 is:
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12π square units
For ellipse x²/a² + y²/b² = 1, area = πab. Here a² = 9 so a = 3, and b² = 16 so b = 4. Area = π × 3 × 4 = 12π square units.
Find the area bounded by y = x³, x-axis, between x = 0 and x = 2.
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4 square units
Area = ∫₀² x³ dx = [x⁴/4]₀² = 16/4 - 0 = 4 square units. The antiderivative of x³ is x⁴/4.
The correct integral setup for finding the area of region bounded by x² = 4y, y = 1, y = 4, and y-axis is:
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∫₁⁴ 2√y dy
From x² = 4y, we get x = 2√y (taking positive root in first quadrant). Area = ∫₁⁴ x dy = ∫₁⁴ 2√y dy.
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