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Determine the area of the region below the parametric curve given by the set of parametric equations. For each problem you may assume that each curve traces out exactly once from right to left for the given range of t. For these problems you should only use the given parametric equations to determine the answer. 1.$x=t^2+5t-1y=40-t^2-2\le t\le 5$ 2.$x=3co{s}^2(t)—si{n}^2(t)y=6+cos(t)-\frac{\pi }{2}\ne t\le 0$ 3.$x=e^{\frac{1}{4}t}—2y=4+e^{\frac{1}{4t}}—e^{\frac{1}{4}t}-6\le t\le 1$ 

Write parametric, equations for the given curve for the given definitions $y=-4x+1$ a) $x=t$ b) $x=\frac{t}{2}$ c) $x=-4t$

Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique. The line segment starting at $P(-1,-3)\text{and ending at}Q(6,-16)$

What is a parametric curve? $x(t)=t3$
$y(t)=t$

Consider the helix represented investigation by the vector-valued function $r(t)=<2\cos t,2\sin t,t>$. Solve for t in the relationship derived in part (a), and substitute the result into the original set of parametric equations. This yields a parametrization of the curve in terms of the arc length parameter s.

Verify that $\Vert r^t(s)\Vert =1$.Consider the helix represented investigation by the vector-valued function $r(t)=<2\cos t,2\sin t,t>$.

That parametric equations contain more information than just the shape of the curve. Write a short paragraph explaining this statement. Use the following example and your answers to parts (a) and (b) below in your explanation. The position of a particle is given by the parametric equations $x=\sin (t)andy=\cos (t)$ where t represents time. We know that the shape of the path of the particle is a circle. a) How long does it take the particle to go once around the circle? Find parametric equations if the particle moves twice as fast around the circle. b) Does the particle travel clockwise or counterclockwise around the circle? Find parametric equations if the particle moves in the opposite direction around the circle.

Explain and solve how the rectangular equation $y=5x$ can have infinitely many sets of parametric equations.

Write the parametric equations $x=3t-1,y=3-3t$ as a function of x in the Cartesian form. $y=?$

At what value of t does the curve $x=2t-3t^2,y=t^2-3t$ have a vertical tangent?