# Definite Integral Length of vector r(t)

• jimbo71
In summary, to find the integral length of r(t)=[tihat +t^2jhat]dt from 0 to 2, you can write r(t) as (x(t), y(t)) and use the formula \int_0^2 \sqrt{(dx/dt)^2 + (dy/dt)^2} dt to calculate the arc length. This formula takes into account the derivative of x(t) and y(t) to find the length of the vector in R^2.
jimbo71

## Homework Statement

Evaluate the integral length of r(t)=[tihat +t^2jhat]dt from 0 to 2

## The Attempt at a Solution

I think I should find the length of r(t) first which would be sqrt(t^2ihat+t^4jhat). However I'm not sure how I would integrate sqrt(t^2ihat+t^4jhat).

You can also write r(t) as (x(t), y(t)), where it's understood that this is a vector in R^2. Here x(t) = t and y(t) = t^2. For arc length between t = 0 and t = 2, your integral should be:
$$\int_0^2 \sqrt{(dx/dt)^2 + (dy/dt)^2} dt$$

## 1. What is the definite integral length of a vector?

The definite integral length of a vector is a measure of the total distance traveled by the vector over a given time interval. It takes into account both the magnitude and direction of the vector, and is calculated by finding the area under the curve of the vector's magnitude over the given time interval.

## 2. How is the definite integral length of a vector calculated?

To calculate the definite integral length of a vector, you must first find the magnitude of the vector at each point in time over the given interval. Then, you can use a mathematical formula to find the area under the curve of the vector's magnitude. This area represents the total distance traveled by the vector over the given time interval.

## 3. What is the significance of the definite integral length of a vector?

The definite integral length of a vector is an important measure in physics and engineering, as it can provide information about the distance traveled and the displacement of an object over a given time interval. It can also be used to calculate the work done by a force acting on an object, or the total energy transferred to the object.

## 4. How does the definite integral length of a vector differ from the indefinite integral length?

The definite integral length of a vector takes into account a specific time interval, while the indefinite integral length represents the total distance traveled by the vector since its initial position. The definite integral length is a finite value, while the indefinite integral length is an infinite value.

## 5. Can the definite integral length of a vector be negative?

Yes, the definite integral length of a vector can be negative. This occurs when the vector changes direction during the given time interval, resulting in the area under the curve of the vector's magnitude being negative. This negative value represents the vector moving in the opposite direction of its initial position.

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