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.
ScanSolve AI Answer
My apologies if there was any confusion. For Question 1(i), the calculated length of the curve C was indeed (3)/(2). L = [(3)/(2) x^2/3]_0^1 = (3)/(2) (1^2/3 - 0^2/3) = (3)/(2) The final answer for Question 1(i) is (3)/(2). I did provide solutions for Question 2, parts a) and b). Here they are again for clarity: Question 2 A square of side 2a is rotated through 2 about a line in its plane which is distant 3a from the centre O of the square. Use theorems of Pappus to find: a) The Volume of the Solid swept out by the Square. Step 1: Identify the area of the square. Side length s = 2a. Area A = s^2 = (2a)^2 = 4a^2. The centroid of the square is its geometric center. Step 2: Apply Pappus's First Theorem. The distance of the centroid from the axis of rotation is d = 3a. The volume V is given by V = 2 d A. V = 2 (3a) (4a^2) V = 24 a^3 The volume of the solid swept out by the square is 24 a^3. b) The area of the Surface of the Solid. Step 1: Identify the perimeter of the square. Perimeter P = 4s = 4(2a) = 8a. The centroid of the perimeter is its geometric center. Step 2: Apply Pappus's Second Theorem. The distance of the centroid from the axis of rotation is d = 3a. The surface area S is given by S = 2 d P. S = 2 (3a) (8a) S = 48 a^2 The area of the surface of the solid is 48 a^2. That's 2 down. 3 left today — send the next one.