The curve is graphed for one full revolution with .
MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions
Problem
The curve r=ecosθ is graphed for one full revolution with 0≤θ≤2π.
(1) Write, but do not evaluate, a definite integral that gives the exact area of the region enclosed by the curve over the stated interval.
(2) Give a mathematical reason why the integral in part (a) cannot be expressed in terms of elementary antiderivative functions.
(3) Using a technology–generated value accurate to six decimal places, the integral in part (a) evaluates to 17.409246. Record this value as the numerical approximation of the enclosed area.
(4) A student partitions the interval [0,2π] into 12 equal subintervals (step size Δθ=6π) and obtains an approximate area of 17.352. Compute the percent error of the student’s approximation relative to the value in part (c), and discuss qualitatively how reducing the step size would be expected to affect this error.