Problem 5 - Calculus Early Transcendentals: Pearson New International Edition

Question

Evaluate each line integral. \(\int_{C}(2 x+9 z) d s ; C\) is the curve \(x=t, y=t^{2}, z=t^{3}\), \(0 \leq t \leq 1\).

Step-by-Step Solution

Answer

The line integral requires numerical computation.

Step 1: Understand the Parameterization

The curve is parameterized by the vector function \(\mathbf{r}(t) = (t, t^2, t^3)\). This means for any point on the curve, \(x = t\), \(y = t^2\), and \(z = t^3\).

Step 2: Express the Integrand in Terms of t

Replace \(x\) and \(z\) in the integrand with their respective parameterized equations: \(2x + 9z = 2t + 9t^3\). Thus, the integrand expressed in terms of \(t\) is \(2t + 9t^3\).

Step 3: Find the Derivative of r(t)

The derivative of \(\mathbf{r}(t)\) with respect to \(t\) is \(\mathbf{r}'(t) = \left(\frac{dx}{dt}, \frac{dy}{dt}, \frac{dz}{dt}\right) = (1, 2t, 3t^2)\).

Step 4: Calculate the Magnitude of the Derivative

The magnitude of the derivative \(\mathbf{r}'(t)\) is \(\|\mathbf{r}'(t)\| = \sqrt{1^2 + (2t)^2 + (3t^2)^2} = \sqrt{1 + 4t^2 + 9t^4}\).

Step 5: Set Up the Integral

Now, set up the integral \(\int_C (2x + 9z) \, ds\) in terms of \(t\): \[ \int_0^1 (2t + 9t^3) \sqrt{1 + 4t^2 + 9t^4} \, dt. \]

Step 6: Simplify and Evaluate the Integral

Unfortunately, this integral does not have a simple analytical solution. You may use numerical integration methods or computational software to approximate it.

Key Concepts

Parameterization of Curves

Parameterization is a fundamental concept when dealing with line integrals, especially in the context of curves. Here, we effectively describe a curve using a parameter, usually denoted as \( t \). In this exercise, the parameterization is given by \( \mathbf{r}(t) = (t, t^2, t^3) \). This means for any point on the curve:

• \( x = t \)
• \( y = t^2 \)
• \( z = t^3 \)

These equations allow us to convert the problem of integrating along a curve in three-dimensional space into a more manageable one-dimensional problem with respect to \( t \). The limits of \( t \) are from 0 to 1, delineating the segment of the curve we are interested in. This conversion plays a crucial part in simplifying the evaluation of line integrals.

Vector Calculus

Vector calculus provides the tools necessary for analyzing and understanding properties of curves and surfaces in multi-dimensional spaces. It encompasses operations like differentiation and integration of vector fields. In this context, we're using vector calculus to process the parameterized curve \( \mathbf{r}(t) = (t, t^2, t^3) \). By taking the derivative \( \mathbf{r}'(t) \), which gives us \( (1, 2t, 3t^2) \), we gain vital information about the curve's direction changes.

• Derivative \( \mathbf{r}'(t) \) helps in computing the curve's length differentials, expressed as \( ds \).
• Magnitude of the derivative provides the rate of change in different coordinates of \( \mathbf{r}(t) \).

This understanding is pivotal, enabling us to formulate the line integral in terms of \( t \), integrating the function over the given path.

Integral Calculus

Integral calculus is key to evaluating line integrals. It involves finding the integral of a function, which is often related to areas under curves, cumulative quantities, or total changes. For line integrals, we apply integral calculus to calculate quantities accumulated along a curve. In this exercise:

• We replaced \( x \) and \( z \) in the integrand with their parameterized equivalents \( 2t + 9t^3 \).
• The integral becomes \( \int_0^1 (2t + 9t^3) \sqrt{1 + 4t^2 + 9t^4} \, dt \).

Though the integral seems complex, integral calculus techniques such as substitution or integration by parts may be explored for simpler forms. In scenarios where these do not simplify the integral, numerical methods can be employed, highlighting integral calculus's flexibility.

Numerical Integration

Numerical integration becomes an essential tool when evaluating integrals that do not have analytical solutions. These techniques approximate the value of integrals, providing solutions in situations where symbolic integration is infeasible. In our exercise, the integral \( \int_0^1 (2t + 9t^3) \sqrt{1 + 4t^2 + 9t^4} \, dt \) appears challenging to simplify analytically.

• Methods such as Simpson's Rule, Trapezoidal Rule, or Gaussian Quadrature can be employed.
• Computational software like Mathematica or MATLAB is often used for precise approximations.

These numerical techniques bridge the gap between complex mathematical problems and practical solutions, enabling us to approximate integrals efficiently in real-world applications.