Application of Integrals – important questions for Boards 2027
CBSE Class 12 Maths
Find the total area between y = x − 1 and the x-axis from x = 0 to x = 2 by adding the absolute areas on either side of the x-axis.
Answer
∫₀¹ (1 − x) dx + ∫₁² (x − 1) dx = 1/2 + 1/2 ∴ Total area = 1 square unit
Using horizontal strips, find the area bounded by x = y², the y-axis, y = 0 and y = 2.
Answer
A = ∫₀² x dy = ∫₀² y² dy = [y³/3]₀² ∴ A = 8/3 square units
Find the area under y = x between x = 0 and x = 2 by expressing the definite integral as a limit of a sum.
Answer
Δx = 2/n xᵣ = 2r/n, r = 1, 2, …, n Area = limₙ→∞ Σᵣ₌₁ⁿ f(xᵣ)Δx = limₙ→∞ Σᵣ₌₁ⁿ (2r/n)(2/n) = limₙ→∞ (4/n²) · n(n + 1)/2 = limₙ→∞ 2(n + 1)/n ∴ Area = 2 square units
Find the geometric area bounded by y = −x, the x-axis, x = 0 and x = 2. Evaluate the definite integral and use its absolute value as the area.
Answer
∫₀² (−x) dx = [−x²/2]₀² = −2 Area = |−2| ∴ Area = 2 square units
Find the area under y = 4 − x² and above the x-axis between x = 0 and x = 2 by evaluating the definite integral of the curve over that interval.
Answer
dA = y dx = (4 − x²) dx A = ∫₀² (4 − x²) dx = [4x − x³/3]₀² = (8 − 8/3) − 0 ∴ A = 16/3 square units
The circle x² + y² = a² is symmetric about both coordinate axes. If the area in its first quadrant is Q, which expression gives the area enclosed by the whole circle? (A) Q (B) 2Q (C) 3Q (D) 4Q
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Find the area enclosed by the ellipse x²/9 + y²/4 = 1.
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For y = x² on 0 ≤ x ≤ 2, write the area dA of a thin vertical strip and form the definite integral that adds all such strips. Evaluate the integral.
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Using integration, find the total area bounded by the line y = 3x + 2, the x-axis, and the ordinates x = −1 and x = 1. Split the integral at the point where the line meets the x-axis.
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Evaluate the total area bounded by y = cos x and the x-axis between x = 0 and x = 2π, splitting the interval at the zeros of cos x.
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5 more probable questions in this chapter, every one with a full answer.
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