Question

Evaluate the line integral along the given path. \(\int_{C} 8 x y z d s\) \(C: \mathbf{r}(t)=12 t \mathbf{i}+5 t \mathbf{j}+3 \mathbf{k}\) \(\quad 0 \leq t \leq 2\)

Short answer

The result of the line integral along the path \(C\) is 3840.

Step-by-step solution

Formulate the Scalar Function in Terms of t

The line integral is over \(ds\), which means that we integrate over the parametrized curve \(C\). So we need to express the scalar function \(8xyz\) in terms of \(t\). This can be done by substituting \(x\), \(y\), and \(z\) from the parametrization of \(C\) into the scalar function. We get \(8 \cdot (12t) \cdot (5t) \cdot 3 = 1440 t^2\)

Calculate the Integral

Now we integrate this new function with respect to \(t\) from 0 to 2. The result of the integration is \(\int_{0}^{2} 1440 t^2 dt = \left. 1440 \cdot \frac{1}{3} t^3 \right|_0^2 = 2^3 \cdot 480 - 0^3 \cdot 480 = 3840\)

Final Evaluation

The result of the integral is a numerical value, which gives us the result of the line integral along the path \(C\).

Key concepts

Understanding Parametric Equations

When we talk about parametric equations, we are referring to a set of equations that express a set of quantities as explicit functions of a number of independent variables, known as parameters. In the context of calculus, these are particularly useful when defining curves in two or more dimensions.

For instance, in the exercise given, the curve C is represented by parametric equations through the vector function \( \mathbf{r}(t)=12 t \mathbf{i}+5 t \mathbf{j}+3 \mathbf{k} \), where \( t \) is the parameter that varies over the interval [0, 2]. Here, \( x(t), y(t), \text{ and } z(t) \) are presented as \( 12t, 5t, \text{ and } 3 \) respectively, with \( t \) taking on values from 0 to 2. These equations offer a way to trace the path of the curve as \( t \) changes.

When solving line integrals involving parametric equations, the first step is often to express the integrand in terms of the parameter, as shown in our exercise. This conversion simplifies the integral into one that can be evaluated using standard techniques of integration.

Scalar Functions in Calculus

A scalar function in calculus is a function that takes one or more variables and returns a single scalar value. This contrasts with vector functions, which return vectors. Scalar functions are fundamental in the field of calculus because they are the building blocks of many types of integrals, including the line integral.

In our exercise, the scalar function is \( 8xyz \), which represents a quantity that depends on three variables: \( x, y, \text{ and } z \). Converting this function in terms of the parameter \( t \) allows us to understand how this scalar quantity changes along the curve C.

Applying the parametric representations of \( x, y, \text{ and } z \) into the scalar function gives us a new scalar function in terms of \( t \): \(8 \cdot (12t) \cdot (5t) \cdot 3 = 1440 t^2\). Now, instead of dealing with three variables, we have a single-variable function, which is crucial for performing the integration step.

Integration of Functions

The integration of functions is a fundamental concept in calculus that involves finding the accumulated quantity represented by the function over a certain interval. In the case of line integrals, we are integrating along a path or curve, which gives us the total accumulation of some scalar quantity across that curve.

In the provided exercise, once we have expressed the scalar integrand in terms of the parameter t, we perform definite integration over the interval from 0 to 2. We calculate the integral \( \int_{0}^{2} 1440 t^2 dt \), which involves finding the antiderivative of \( 1440 t^2 \) and evaluating it at the endpoints of the interval. The step-by-step solution offered simplifies this process by performing the integration and providing the final numeric result,

The technique shown in this example is a vital tool in all fields that use calculus, from physics to engineering, as it enables the calculation of various physical quantities such as work, mass, and charge over a path or within a certain region of space.