To find the coordinates of the centroid of the given curve defined by the parametric equations x = t sin(t) and y = t cos(t) + t sin(t) for 0 t
We can find the coordinates of the centroid (C) of a curve given in parametric form using the following formulas:
- For the x-coordinate: C_x = rac{1}{L} imes ext{integral}(x(t) imes rac{dy}{dt} imes dt)
- For the y-coordinate: C_y = rac{1}{L} imes ext{integral}(y(t) imes rac{dy}{dt} imes dt)
Where:
- L is the length of the curve, calculated as: L = ext{integral}( rac{dy}{dt}) imes dt
To get the derivatives:
dx/dt = sin(t) + t cos(t)
dy/dt = cos(t) + sin(t) + t cos(t)
Now, substituting these into the formulas and evaluating from t = 0 to t = π/2, we’re able to calculate the centroid coordinates (C_x, C_y).
This process may require numerical approximation or symbolic integration depending on the complexity of the functions involved.