Question

Compute the total mass of a wire bent in a quarter circle with parametric equations: x=9cost, y=9sint, 0≤t≤π2 and density function rho(x,y)=x²+y².

Answer 1

To find the total mass of the wire bent in a quarter circle with the given parametric equations and density function, we calculate the line integral of the density function along the curve using the differential arc length element.

Step-by-step explanation:

The student is asking to calculate the total mass of a wire bent in a quarter circle defined by parametric equations x=9cos(t) and y=9sin(t) for 0≤t≤π/2, with a given density function ρ(x,y)=x²+y².

We can find the mass of the wire by integrating the density function along the curve. Since the wire is described parametrically, we'll use the line integral of a scalar field formula:

∫ ρ(x,y) ds

Where ds is the differential arc length element. We can find ds using the derivatives of the parametric equations with respect to t:

ds = √((dx/dt)² + (dy/dt)²) dt

By substituting the density function and calculating the integral, we'll obtain the total mass of the wire which is the solution to the problem.