Index Numbers

Answer

Area = ∫ᵇₐ f(x) dx This formula gives the area bounded by: • The curve y = f(x) • The X-axis • The vertical lines x = a and x = b Note: The function must be continuous in [a, b] and f(x) ≥ 0 for thi…

Answer

Area = ∫ᵈᶜ x dy = ∫ᵈᶜ g(y) dy This formula is used when: • The curve is expressed as x in terms of y • The region is bounded by horizontal lines • Integration is performed with respect to y Example:…

Answer

Step 1: Set up the integral Area = ∫³₁ y dx = ∫³₁ x² dx Step 2: Integrate = [x³/3]³₁ Step 3: Apply limits = 3³/3 - 1³/3 = 27/3 - 1/3 = 9 - 1/3 = 26/3 sq. units Answer: 26/3 square units…

Answer

When f(x) < 0, the definite integral gives a negative value, but area is always positive. Solution: Take the absolute value Area = |∫ᵇₐ f(x) dx| Example: y = -x², x = 1 to x = 2 ∫²₁ (-x²) dx = [-x³/…

Answer

Area = |∫ᵇₐ [f(x) - g(x)] dx| Step-by-step approach: 1. Find intersection points by solving f(x) = g(x) 2. Determine which curve is above the other 3. Set up integral as (upper curve - lower curve) 4…

Answer

Step 1: Find intersection points From y² = 9x and x² = 9y Substitute y = x²/9 into first equation: (x²/9)² = 9x → x⁴/81 = 9x → x⁴ = 729x x(x³ - 729) = 0 → x = 0 or x = 9 Points: (0,0) and (9,9) Step …

Answer

Since sin x changes sign, we split the integral: Step 1: Identify regions • From 0 to π: sin x ≥ 0 (above X-axis) • From π to 2π: sin x ≤ 0 (below X-axis) Step 2: Calculate each area A₁ = ∫ᵖ₀ sin x …

Answer

Step 1: Use symmetry Total area = 4 × (area in first quadrant) Step 2: Express y in terms of x From ellipse equation: y = (b/a)√(a² - x²) Limits: x = 0 to x = a Step 3: Set up integral Area = 4∫ᵃ₀ (…