Applications of Integration — Flashcards
Tamil Nadu Board · Class 12 · Mathematics
25 flashcards for Applications of Integration (Tamil Nadu Board Class 12 Mathematics) to test yourself on key terms and facts.
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Evaluate $\int_0^3 (3x^2 - 4x + 5) dx$ using the Fundamental Theorem of Integral Calculus.
Answer
Step 1: Find the antiderivative of each term: - ∫3x² dx = x³ - ∫(-4x) dx = -2x² - ∫5 dx = 5x So F(x) = x³ - 2x² + 5x Step 2: Apply F(b) - F(a) where a=0, b=3: F(3) = 27 - 2(9) + 5(3) = 27 - 18 + 15 =…
Solve: $\int_0^1 x e^{2x} dx$ using integration by parts.
Answer
Step 1: Use integration by parts formula: ∫u dv = uv - ∫v du Let u = x, dv = e^(2x) dx Then du = dx, v = e^(2x)/2 Step 2: Apply the formula: ∫x e^(2x) dx = (x · e^(2x)/2) - ∫(e^(2x)/2) dx = xe^(2x)/2…
When do you use the property $\int_0^a f(x) dx = \int_0^a f(a-x) dx$? Give an example application.
Answer
Use this property when the integrand has a symmetric form that simplifies after substitution. Example: Evaluate ∫₀^(π/4) log(1 + tan x) dx Step 1: Using the property with a = π/4: ∫₀^(π/4) log(1 + t…
Calculate the area of the region bounded by the parabola $y = x^2$, the x-axis, and the lines $x = 1$ and $x = 2$.
Answer
Step 1: Identify the region. The parabola y = x² lies above the x-axis between x = 1 and x = 2. Step 2: Set up the area integral: Since y ≥ 0 in this region: A = ∫₁² y dx = ∫₁² x² dx Step 3: Find th…
Find the area of the region bounded between the curves $y = x^2$ and $y = x$.
Answer
Step 1: Find intersection points by solving x² = x: x² - x = 0 x(x - 1) = 0 So x = 0 and x = 1 Step 2: Determine which curve is on top: At x = 0.5: y = x² gives 0.25, y = x gives 0.5 So y = x is abov…
Evaluate $\int_{-2}^{2} x^3 dx$ using the property of odd functions.
Answer
Step 1: Check if f(x) = x³ is an odd function: f(-x) = (-x)³ = -x³ = -f(x) ✓ Since f(-x) = -f(x), the function is odd. Step 2: Apply the odd function property: For odd functions: ∫₋ₐᵃ f(x) dx = 0 St…
Formula for evaluating $\int_0^{\pi/2} \sin^n x \, dx$ when n is a positive integer. What does it equal when n = 4?
Answer
Reduction Formula for ∫₀^(π/2) sin^n x dx: When n is even (n = 2, 4, 6, ...): ∫₀^(π/2) sin^n x dx = ((n-1)/n) × ((n-3)/(n-2)) × ... × (1/2) × (π/2) For n = 4: Step 1: Apply the formula: ∫₀^(π/2) sin…
Use substitution to evaluate $\int_0^{\pi/3} \frac{\sec x \tan x}{1 + \sec^2 x} dx$.
Answer
Step 1: Choose substitution: Let u = sec x Then du = sec x tan x dx Step 2: Change the limits: When x = 0: u = sec(0) = 1 When x = π/3: u = sec(π/3) = 2 Step 3: Substitute in the integral: ∫₀^(π/3) …
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