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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Okay 🌹CONFIDENTIALITY, let's finish these up. Q8. Find the volume generated when y = x^2 is rotated about the x-axis from x = 0 to x = 1. Step 1: Use the disk method formula for the volume of revolution about the x-axis: V = _a^b [f(x)]^2 \, dx. Here, f(x) = x^2, a = 0, and b = 1. V = _0^1 (x^2)^2 \, dx = _0^1 x^4 \, dx Step 2: Find the antiderivative of x^4. x^4 \, dx = x^4+14+1 = (x^5)/(5) Step 3: Evaluate the antiderivative at the limits of integration and multiply by . V = [(x^5)/(5)]_0^1 = ((1^5)/(5) - (0^5)/(5)) Step 4: Calculate the numerical value. V = ((1)/(5) - 0) = ()/(5) The final answer is ()/(5) cubic units. Q9. Find the work done in stretching a spring if force is given by F = 5x, from x = 0 to x = 2. Step 1: The work done by a variable force is given by the definite integral of the force function with respect to displacement. W = _a^b F(x) \, dx Here, F(x) = 5x, a = 0, and b = 2. W = _0^2 5x \, dx Step 2: Find the antiderivative of 5x. 5x \, dx = 5 · x^1+11+1 = 5 · (x^2)/(2) = (5)/(2)x^2 Step 3: Evaluate the antiderivative at the limits of integration. W = [(5)/(2)x^2]_0^2 = ((5)/(2)(2)^2) - ((5)/(2)(0)^2) Step 4: Calculate the numerical value. W = ((5)/(2) · 4) - 0 = 10 The final answer is 10 units of work. Q10. Find the work done by a force F = 3x^2 from x = 1 to x = 3. Step 1: Set up the definite integral for the work done. W = _a^b F(x) \, dx Here, F(x) = 3x^2, a = 1, and b = 3. W = _1^3 3x^2 \, dx Step 2: Find the antiderivative of 3x^2. 3x^2 \, dx = 3 · x^2+12+1 = 3 · (x^3)/(3) = x^3 Step 3: Evaluate the antiderivative at the limits of integration. W = [x^3]_1^3 = (3)^3 - (1)^3 Step 4: Calculate the numerical value. W = 27 - 1 = 26 The final answer is 26 units of work. Q11. Find the average value of f(x) = x^2 on [0,3]. Step 1: Use the formula for the average value of a function f(x) on an interval [a, b]: f_avg = (1)/(b-a) _a^b f(x) \, dx. Here, f(x) = x^2, a = 0, and b = 3. f_avg = (1)/(3-0) _0^3 x^2 \, dx = (1)/(3) _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 definite integral. _0^3 x^2 \, dx = [(x^3)/(3)]_0^3 = ((3^3)/(3)) - ((0^3)/(3)) = (27)/(3) - 0 = 9 Step 4: Multiply by (1)/(b-a). f_avg = (1)/(3) · 9 = 3 The final answer is 3. Last free one today — make it count tomorrow, or type /upgrade for unlimited.