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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Right 🌹CONFIDENTIALITY, let's tackle these area problems. Q1. Find the area under the curve y = x^2 from x = 0 to x = 3. Step 1: Set up the definite integral for the area. A = _0^3 x^2 \, dx Step 2: Find the antiderivative of x^2. x^2 \, dx = x^2+12+1 = (x^3)/(3) Step 3: Evaluate the antiderivative at the upper and lower limits using the Fundamental Theorem of Calculus. A = [(x^3)/(3)]_0^3 = ((3^3)/(3)) - ((0^3)/(3)) Step 4: Calculate the numerical value. A = (27)/(3) - 0 = 9 The final answer is 9. Q2. Find the area under y = 2x + 1 from x = 1 to x = 4. Step 1: Set up the definite integral for the area. A = _1^4 (2x + 1) \, dx Step 2: Find the antiderivative of 2x + 1. (2x + 1) \, dx = 2 · (x^2)/(2) + x = x^2 + x Step 3: Evaluate the antiderivative at the upper and lower limits. A = [x^2 + x]_1^4 = ((4)^2 + 4) - ((1)^2 + 1) Step 4: Calculate the numerical value. A = (16 + 4) - (1 + 1) = 20 - 2 = 18 The final answer is 18. Q3. Find the area between y = x^2 and y = x from x = 0 to x = 1. Step 1: Determine which function is greater over the interval [0, 1]. For x (0, 1), x > x^2. For example, if x=0.5, then x=0.5 and x^2=0.25. So y=x is the upper curve and y=x^2 is the lower curve. Step 2: Set up the definite integral for the area between the curves. A = _0^1 (x - x^2) \, dx Step 3: Find the antiderivative of x - x^2. (x - x^2) \, dx = (x^2)/(2) - (x^3)/(3) Step 4: Evaluate the antiderivative at the upper and lower limits. A = [(x^2)/(2) - (x^3)/(3)]_0^1 = ((1^2)/(2) - (1^3)/(3)) - ((0^2)/(2) - (0^3)/(3)) Step 5: Calculate the numerical value. A = ((1)/(2) - (1)/(3)) - (0 - 0) = (3)/(6) - (2)/(6) = (1)/(6) The final answer is (1)/(6). 3 done, 2 left today. You're making progress.