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
Step 1: Parameterize the curve C. The curve C is a circle of radius R=4 centered at the origin. We can parameterize it using t as the angle: x(t) = 4 t y(t) = 4 t for 0 t 2. So, the position vector is r(t) = 4 t, 4 t . Step 2: Calculate r'(t) and its magnitude |r'(t)|. First, find the derivative of the parameterization: r'(t) = (d)/(dt)(4 t), (d)/(dt)(4 t) = -4 t, 4 t Next, find the magnitude of r'(t): |r'(t)| = sqrt((-4 t)^2 + (4 t)^2) |r'(t)| = sqrt(16 ^2 t + 16 ^2 t) |r'(t)| = sqrt(16(^2 t + ^2 t)) Using the identity ^2 t + ^2 t = 1: |r'(t)| = sqrt(16(1)) = sqrt(16) = 4 So, ds = |r'(t)| \, dt = 4 \, dt. Step 3: Express the integrand f(x,y) = x^2+y^2 in terms of t. Substitute x=4 t and y=4 t into the integrand: f(x(t), y(t)) = (4 t)^2 + (4 t)^2 f(x(t), y(t)) = 16 ^2 t + 16 ^2 t f(x(t), y(t)) = 16(^2 t + ^2 t) Using the identity ^2 t + ^2 t = 1: f(x(t), y(t)) = 16(1) = 16 Step 4: Set up and evaluate the line integral. The line integral is given by _C f(x,y) \, ds = _a^b f(x(t), y(t)) |r'(t)| \, dt. Substitute the expressions found in the previous steps: _C (x^2 + y^2) \, ds = _0^2 (16)(4) \, dt _0^2 64 \, dt Now, evaluate the definite integral: [64t]_0^2 = 64(2) - 64(0) = 128 The value of the integral is 128. That's 2 down. 3 left today — send the next one.