Time Series — Practice Quiz
Maharashtra Board · Class 12 · Mathematics & Statistics -Commerce
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Quick Quiz: Time Series
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Evaluate ∫₁³ (2x + 1) dx using the fundamental theorem of calculus.
Calculate ∫₀² x² dx.
If ∫₋₂² f(x) dx where f(x) = x³, what is the value?
Evaluate ∫₀^(π/2) sin x dx.
Sample Questions
Which of the following properties of definite integrals are correct?
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∫ₐᵃ f(x) dx = 0, ∫ᵇₐ f(x) dx = -∫ₐᵇ f(x) dx, ∫ᵇₐ f(x) dx = ∫ᶜₐ f(x) dx + ∫ᵇᶜ f(x) dx, ∫ᵇₐ f(x) dx = ∫ᵇₐ f(t) dt
All four are fundamental properties: (1) Same limits give zero, (2) Reversing limits changes sign, (3) Splitting interval property, (4) Variable independence property
Which of the following are correct for even functions?
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f(-x) = f(x), ∫₋ₐᵃ f(x) dx = 2∫₀ᵃ f(x) dx, Graph is symmetric about y-axis
For even functions: f(-x) = f(x), symmetric about y-axis, and integral over symmetric interval equals twice the integral from 0 to a. The property f(-x) = -f(x) applies to odd functions.
Calculate ∫₁⁴ (1/√x) dx.
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2
Step 1: Rewrite 1/√x as x^(-1/2) Step 2: Antiderivative is x^(1/2)/(1/2) = 2√x Step 3: Apply limits: [2√x]₁⁴ = 2√4 - 2√1 = 2(2) - 2(1) = 4 - 2 = 2
If ∫₀¹ f(x) dx = 3 and ∫₁² f(x) dx = 5, find ∫₀² f(x) dx.
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8
Using the additive property of definite integrals: ∫₀² f(x) dx = ∫₀¹ f(x) dx + ∫₁² f(x) dx = 3 + 5 = 8
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