Question

Sketch the plane curve and find its length over the given interval. $$ \mathbf{r}(t)=t^{3} \mathbf{i}+t^{2} \mathbf{j}, \quad[0,2] $$

Short answer

The curve is sketched by plotting points \((t^{3}, t^{2})\) for \( t \) in [0,2]. The length of the curve from 0 to 2 is obtained by integrating the magnitude of the derivative of \( \mathbf{r}(t) \), which is the integral from 0 to 2 of \( | \mathbf{r}'(t) | = \sqrt{9t^{4} + 4t^{2}} dt \).

Step-by-step solution

Sketch the Curve

Firstly, sketch the curve by varying \( t \) in the interval [0,2] and plotting the points \((t^{3}, t^{2})\). You may notice that as \( t \) increases, the x and y coordinates smoothly increase as well, meaning the curve moves in one direction without any sharp turns or loops.

Find the Derivative of r(t)

Now, we need to find the derivative of \( \mathbf{r}(t) \) which will be used to calculate the length of the curve. This can be found by differentiating each component of \(t^{3}\) and \(t^{2}\) to get: \[ \mathbf{r}'(t) = 3t^{2}\mathbf{i} + 2t\mathbf{j} \].

Calculate the Magnitude of the Derivative

Calculate the magnitude of the derivative, which is the square root of the sum of the squares of the components:\[ |\mathbf{r}'(t)| = \sqrt{ (3t^{2})^{2} + (2t)^{2} } = \sqrt{9t^{4} + 4t^{2}}.\]This needs to be integrated from 0 to 2 for calculating the length.

Find the Length of the Curve

Find the length of the curve by integrating the magnitude of the derivative from 0 to 2. This can be evaluated as:\[L = \int_{0}^{2} \sqrt{9t^{4} + 4t^{2}} dt.\]Perform the integral to find the length over the given interval, which requires techniques of integration.

Key concepts

Sketching Curves in Calculus

Sketching curves is often the first step in visualizing and understanding the behavior of functions in calculus. When dealing with a parametric equation, like \( \mathbf{r}(t)=t^{3} \mathbf{i}+t^{2} \mathbf{j} \), the process involves plotting points for various values of the parameter \( t \) and connecting these points to form a curve.

When sketching a parametric curve, it's important to consider the direction of motion as \( t \) increases. For the equation provided, the curve generated by plotting points \( (t^3, t^2) \) on a coordinate plane shows both the x and y components increasing as \( t \) goes from 0 to 2, creating a graph that smoothly rises without sharp turns or loops. This information is not only helpful for visual learners but also gives a sense of the curve's geometry which is essential when calculating length or analyzing the curve's properties.

Differentiation of Parametric Equations

Differentiation plays a crucial role in finding the properties of a parametric curve. To differentiate parametric equations, you independently find the derivatives of the individual component functions. For instance, in the equation \( \mathbf{r}(t)=t^{3} \mathbf{i}+t^{2} \mathbf{j} \), separate differentiation of \( t^3 \) and \( t^2 \) results in \( 3t^2 \) and \( 2t \), respectively.

Combining these derivatives, we get the derivative of the vector-valued function \( \mathbf{r}'(t)=3t^{2} \mathbf{i}+2t \mathbf{j} \). This derivative vector gives the rate of change of the position vector \( \mathbf{r}(t) \) with respect to \( t \) and is essential for determining the curve's slope, tangent lines, and calculating the length of the curve.

Arc Length of Curve

The length of a curve in the plane, or arc length, can be computed for parametric curves by integrating the magnitude of the derivative over the desired interval. The magnitude of the derivative, represented by \( |\mathbf{r}'(t)| \), comes from the Pythagorean theorem and accounts for changes in both the x and y directions.

For the given curve \( \mathbf{r}(t)=t^{3} \mathbf{i}+t^{2} \mathbf{j} \), the magnitude is \( \sqrt{(3t^{2})^{2} + (2t)^{2}} \). The arc length \( L \) is then obtained by integrating this magnitude from the starting point \( t=0 \) to the endpoint \( t=2 \), expressed as\[ L = \int_{0}^{2} \sqrt{9t^{4} + 4t^{2}} \,dt \].
To calculate the exact numerical length, one must apply integration techniques, which might involve substitution or trigonometric identities, depending on the complexity of the integrand.