# 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 - 1 y = 40 - t^2 -2 leq t leq 5 2.x = 3cos^2 (t) — sin^2 (t) y = 6 + cos(t) -frac{pi}{2} neq t leq 0 3.x = e^{frac{1}{4} t} —2 y = 4 + e^{frac{1}{4 t}} — e^{frac{1}{4} t} - 6 leq t leq 1

Question
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.$$x = t^2 + 5t - 1 y = 40 - t^2 -2 \leq t \leq 5$$ 2.$$x = 3cos^2 (t) — sin^2 (t) y = 6 + cos(t) -\frac{\pi}{2} \neq t \leq 0$$ 3.$$x = e^{\frac{1}{4} t} —2 y = 4 + e^{\frac{1}{4 t}} — e^{\frac{1}{4} t} - 6 \leq t \leq 1$$

2020-11-07
Given: As per our honor code, we are answering only the first question. The parametric equations are $$x = t^2 + 5t − 1, y = 40 − t^2, − 2 \leq t \neq 5$$. We have to find the area of the region bounded by these parametric equations. Concept Used: If the parametric equations are $$x=f(t), y=g(t)$$ and
$$a \leq t \leq b$$ then the area of the bounded region is given by $$A = int_a^b g(t) f (t) dt$$ Calculating:

### Relevant Questions

1. A curve is given by the following parametric equations. x = 20 cost, y = 10 sint. The parametric equations are used to represent the location of a car going around the racetrack. a) What is the cartesian equation that represents the race track the car is traveling on? b) What parametric equations would we use to make the car go 3 times faster on the same track? c) What parametric equations would we use to make the car go half as fast on the same track? d) What parametric equations and restrictions on t would we use to make the car go clockwise (reverse direction) and only half-way around on an interval of [0, 2?]? e) Convert the cartesian equation you found in part “a” into a polar equation? Plug it into Desmos to check your work. You must solve for “r”, so “r = ?”
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 $$x =\ \sin\ (t)\ and\ y =\ \cos\ (t)$$ 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.
Write a short paragraph explaining this statement. Use the following example and your answers 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. The position of a particle is given by the parametric equations $$x = sin t, y = cos t$$ where 1 represents time. We know that the shape of the path of the particle is a circle.
Parametric equations and a value for the parameter t are given $$x = (60 cos 30^{\circ})t, y = 5 + (60 sin 30^{\circ})t - 16t2, t = 2$$. Find the coordinates of the point on the plane curve described by the parametric equations corresponding to the given value of t.
Working with parametric equations Consider the following parametric equations. a. Eliminate the parameter to obtain an equation in x and y. b. Describe the curve and indicate the positive orientation. $$x=\ -7\ \cos\ 2t,\ y=\ -7\ \sin\ 2t,\ 0\ \leq\ t\ \leq\ \pi$$
Parametric to polar equations Find an equation of the following curve in polar coordinates and describe the curve. $$x = (1 + cos t) cos t, y = (1 + cos t) sin t, 0 \leq t \leq 2\pi$$
Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. $$x=3\ \ln(t),\ y=4t^{\frac{1}{2}},\ z=t^{3},\ (0,\ 4,\ 1)$$
Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. $$x=e^{-8t}\ \cos(8t),\ y=e^{-8t}\ \sin(8t),\ z=e^{-8t},\ (1,\ 0,\ 1)$$
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.
Write a short paragraph explaining this statement. Use the following example and your answers Does the particle travel clockwise or anticlockwise around the circle? Find parametric equations if the particles moves in the opposite direction around the circle. The position of a particle is given by the parametric equations $$x = sin t, y = cos t$$ where 1 represents time. We know that the shape of the path of the particle is a circle.