A particle travels in the plane along the parametric curve C defined by x(t)=t^ 3 -3t

MATHEMATICS · Multi-part · AP · AP Precalculus · Functions with Parameters, Vectors, and Matrices · Functions with Parameters, Vectors, and Matrices

Problem

A particle travels in the plane along the parametric curve C defined by

x(t)=t^{3}-3t, \qquad y(t)=t^{2}-2, \qquad -2\le t\le 2.

The curve intersects the

x

–axis at the two parameter values

t=-\sqrt{2}

and

t=\sqrt{2}

. The region R is bounded by the portion of C between those intersection points and the

x

–axis. (a) Determine the direction of motion (leftward or rightward) of the particle at each of the parameter values

t=-2,-1,1,

and

2

. Justify your conclusions. (b) Identify the complete interval(s) of

t

on which the curve moves from right to left across the

xy

–plane. Explain your reasoning. (c) Without performing any numerical integration, decide whether the single definite integral

\displaystyle \int_{-\sqrt{2}}^{\sqrt{2}} y(t)\,x'(t)\,dt

is positive, negative, or zero. Provide a clear justification based on the orientation of the curve and the sign of the integrand. (d) Describe how the integral in part (c) must be modified, if at all, to obtain the positive area of region R. Explain your answer verbally without computing the value of the area.

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