This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here is a complex math problem involving calculus and algebra: Find the area of the region bounded by the curves y = x² and y = √(x). Step 1: Find the points of intersection of the two curves. Set x² = √(x). Square both sides: (x²)² = (√(x))² x⁴ = x x⁴ - x = 0 x(x³ - 1) = 0 This gives x = 0 or x³ - 1 = 0, which means x³ = 1, so x = 1. The points of intersection are at x = 0 and x = 1. Step 2: Determine which function is greater in the interval [0, 1]. Let's test a value between 0 and 1, for example, x = 0.5. For y = x², y = (0.5)² = 0.25. For y = √(x), y = √(0.5) ≈ 0.707. So, √(x) is greater than x² in the interval [0, 1]. Step 3: Set up the integral for the area. The area A is given by the integral of the upper curve minus the lower curve from the lower bound to the upper bound of intersection. A = ∫ from 0 to 1 of (√(x) - x²) dx Step 4: Evaluate the integral. A = ∫ from 0 to 1 of (x^(1/2) - x²) dx Integrate term by term: A = [ (x^(3/2))/(3/2) - (x³)/3 ] from 0 to 1 A = [ (2/3)x^(3/2) - (1/3)x³ ] from 0 to 1 Step 5: Apply the limits of integration. A = [ (2/3)(1)^(3/2) - (1/3)(1)³ ] - [ (2/3)(0)^(3/2) - (1/3)(0)³ ] A = [ (2/3) - (1/3) ] - [ 0 - 0 ] A = (1/3) The area of the region bounded by the curves y = x² and y = √(x) is (1/3) square units. Send me the next one 📸