Question

Evaluate \(\int_{C}(x+4 \sqrt{y}) d s\) along the given path. \(C:\) line from (0,0) to (3,9)

Short answer

The line integral \(\int_{C}(x+4 \sqrt{y}) d s\) along the line from (0,0) to (3,9) becomes \(\int_{0}^{3}(t+4 \sqrt{3 t^{2}}) \sqrt{1+ 36 t^2} dt\) after substituting the parameters. This integral can be evaluated using basic integral rules or numerical methods.

Step-by-step solution

Parametrize the Line

Let's parametrize the line segment from (0,0) to (3,9). A typical point on the line can be written as \((x,y) = (t, 3t^{2})\), where \(t\) varies from 0 to 3.

Calculate ds

Next, in order to integrate along the line segment, we need to find ds, which is the magnitude of the derivative of the parametric representation, For our parametric equations, we find ds by: \(ds = \sqrt{(dx/dt)^2 + (dy/dt)^2} dt = \sqrt{(1)^2 + (6t)^2} dt = \sqrt{1+36t^{2}} dt.\)

Substitute for x,y and ds in the integral

Substitute \(x = t\), \(y = 3t^2\), and \(ds = \sqrt{1+36t^{2}} dt\) into the integral. The given integral becomes: \(\int_{0}^{3}(t+4 \sqrt{3 t^{2}}) \sqrt{1+ 36 t^2} dt.\)

Evaluate the Integral

The last step is to evaluate the definite integral from 0 to 3. This may involve using basic integral rules or more commonly, numerical methods either by graphing calculator or computer software program such as R or Python for a more accurate result. From here the student will evaluate the integral.

Key concepts

Parametric Equations

When dealing with line integrals in calculus, parametric equations play a fundamental role. These equations express the coordinates of points on a curve as functions of a single parameter, typically denoted as t. For instance, consider the line segment from (0,0) to (3,9). By parametrizing this segment, we describe every point (x,y) along the line as (x(t), y(t)), where (t) is a real number within a specific range to cover the whole segment.

In our example, the parametric equations are (x(t) = t) and (y(t) = 3t^2), which indicates that as (t) changes, the point (x,y) traces out the path from the starting point to the ending point. This set of parametric equations simplifies the process of integration along the curve, making it feasible to perform complex calculations that are otherwise difficult to manage with standard Cartesian coordinates.

Definite Integral

The definite integral is a central concept in calculus, representing the accumulation of quantities as a function changes. In the context of line integrals, the definite integral accumulates a quantity along a path, often represented by a curve or line segment. With parametric equations set, the definite integral transforms the problem into evaluating the integral of a function over a specific interval.

In our problem, after substituting the parametric forms into the integral, we are left with a new integral to evaluate: \[\int_{0}^{3}(t+4 \sqrt{3 t^{2}}) \sqrt{1+ 36 t^2} dt.\]
This is our definite integral, which is to be calculated from the lower limit (t=0) to the upper limit (t=3). The result of this computation will yield the value of the line integral over the curve C.

Numerical Methods

Numerical methods come into play when an integral cannot be solved analytically or when we seek a practical approximation. These methods are algorithms for approximating the solutions of mathematical problems, including definite integrals. With integrals such as \[\int_{0}^{3}(t+4 \sqrt{3 t^{2}}) \sqrt{1+ 36 t^2} dt,\]
which might require complex algebra to solve exactly, numerical approximation can be a valuable tool.

Common numerical methods include the Trapezoidal Rule, Simpson's Rule, and Riemann Sums. Software like R, Python, or even graphing calculators can implement these methods to produce an accurate approximation of definite integrals. In some cases, students are guided to use these tools to obtain their answers, especially when integrals are either very difficult or impossible to solve by hand.