A thin wire is bent into a quarter-circle of radius 4 meters in the first quadrant
MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions
Problem
A thin wire is bent into a quarter-circle of radius 4 meters in the first quadrant, beginning at the point (4,0) and ending at (0,4). Let θ denote the central angle, in radians, measured from the positive x-axis to a typical point on the wire. The wire’s linear mass density, in kilograms per meter of arclength, is modeled by λ(θ)=3+sinθ for 0≤θ≤2π.
(a) Express the differential element of arclength ds along the wire in terms of dθ.
(b) Write, and then evaluate, a definite integral that gives the total mass of the quarter-circular wire. Give your answer to three significant figures.