# Find Distance Covered by Point P on Parametric Curve: 0 to 9

• ILoveBaseball
In summary, the conversation discusses setting up an integral to find the distance traveled by a point on a parametric curve between specified values of t. The correct formula for calculating the length of a curve is provided and it is mentioned that the given curve is actually a straight line. The conversation ends with a thank you for the help.
ILoveBaseball
Consider the parametric curve given by the equations
$$x(t) = t^2+30t-11$$
$$y(t)=t^2+30t+38$$
How many units of distance are covered by the point P(t) = (x(t),y(t)) between t=0, and t=9 ?

well since the bounds are already given (0->9), i just need help on setting up the integral. here's what i done:

$$dx/dt = 2t+30$$

my integral:

$$\int_{0}^{9}(t^2+30t+38)*(2t+30)$$

but i get the incorrect answer when i integral it, can someone help me set it up?

ILoveBaseball said:
Consider the parametric curve given by the equations
$$x(t) = t^2+30t-11$$
$$y(t)=t^2+30t+38$$
How many units of distance are covered by the point P(t) = (x(t),y(t)) between t=0, and t=9 ?

well since the bounds are already given (0->9), i just need help on setting up the integral. here's what i done:

$$dx/dt = 2t+30$$

my integral:

$$\int_{0}^{9}(t^2+30t+38)*(2t+30)$$

but i get the incorrect answer when i integral it, can someone help me set it up?

The distance traveled is

$$\int_{0}^{9}\sqrt{(dx/dt)^2+(dy/dt)^2}dt$$

You either do this integral, or notice that the curve is a straight line.

ehild

What is the formula for calculating curve's length? You have calculated the area under the curve, not the length.

thank you, i got it correct

## 1. What is a parametric curve?

A parametric curve is a mathematical concept that describes a curve or line in terms of two or more functions, also known as parameters, that determine the position of a point on the curve. The values of these parameters change as the point moves along the curve, allowing for a dynamic description of the curve.

## 2. How is the distance covered by a point on a parametric curve calculated?

The distance covered by a point on a parametric curve is calculated by using the distance formula, which takes into account the change in the x and y coordinates of the point as it moves along the curve. This formula is: d = √((x2-x1)^2 + (y2-y1)^2).

## 3. What does the range of 0 to 9 mean in relation to the parametric curve?

The range of 0 to 9 refers to the interval or domain of the parametric curve, which indicates the values of the parameter(s) that will be used to describe the curve. In this case, the range of 0 to 9 means that the point will cover a distance on the curve from the starting point at 0 to the end point at 9.

## 4. How is the distance covered by a point on a parametric curve affected by changes in the parameters?

The distance covered by a point on a parametric curve is directly affected by changes in the parameters. As the parameters change, the position of the point on the curve will also change, resulting in a different distance covered. This is why parametric curves are useful for describing dynamic or changing systems.

## 5. Can a point on a parametric curve cover an infinite distance?

No, a point on a parametric curve cannot cover an infinite distance. The distance covered by the point will always be finite, as it is limited by the range of the parameters and the shape of the curve. However, the distance covered can approach infinity if the parameters continue to increase without bound.

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