(a) Graph the curve traced out by the vector function 𝐫=𝐫(t) (b) Find the indicated arc length.

step 1 Here we have a vector function that we're going to graph and then we're going to find the arc le

(a) Graph the curve traced out by the vector function 𝐫=𝐫(t)...

(a) Graph the curve traced out by the vector function $\mathbf{r}=\mathbf{r}(t)$
(b) Find the indicated arc length.
$$
\mathbf{r}(t)=4 \sin (2 t) \mathbf{i}-6 \cos t \mathbf{j} \text { from } t=0 \text { to } t=\frac{\pi}{2}
$$

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Key Concepts

Arc Length Calculation

Calculating the arc length of a curve involves integrating the magnitude of the derivative of a vector function over the given interval. This process is a fundamental application of integration in vector calculus, connecting the concepts of differentiation and the physical notion of distance along a curve.

Differentiation of Vector Functions

Differentiating a vector function with respect to its parameter yields the velocity vector, which indicates the direction of the curve and the rate at which the position is changing. This concept is also essential in finding other properties of the curve, such as arc length and curvature.

Vector Functions

A vector function describes a curve in space by expressing each coordinate as a function of a parameter. This concept is central in calculus and physics, as it allows one to define paths, motions, and trajectories in two or three dimensions in a compact form.

Parametric Representation of Curves

Parametric representation is a method where each coordinate of a point on a curve is given as an independent function of a parameter. This form provides flexibility in describing curves that may not be easily expressible as functions in the conventional Cartesian coordinate system.

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