91Ó°ÊÓ

A vector integral extends the concept of a typical integral to vector-valued functions. Unlike regular integrals, which deal with scalars, vector integrals focus on vectors composed of several components. Each component is a separate integral. In the presented exercise, the vector function is given as \( \mathbf{F}(t) = t^3 \mathbf{i} + 7 \mathbf{j} + (t+1) \mathbf{k} \).

• The \( \mathbf{i} \) component is \( t^3 \)
• The \( \mathbf{j} \) component is \( 7 \)
• The \( \mathbf{k} \) component is \( t+1 \)

When performing a vector integral, you calculate the integral for each of these components separately and then recombine them.
The final result is a new vector consisting of the integrated components.With vector integrals, keep in mind:

• Break down the vector into its individual scalar component functions.
• Integrate each component independently within the specified limits.
• Combine the results to form the integrated vector result.

Integration of vector functions involves treating each component of a vector function as an independent scalar function. This is a valuable technique in vector calculus as it simplifies complex integrals into manageable parts. Let's illustrate this with the example:
In the vector function \( \mathbf{F}(t) = t^3 \mathbf{i} + 7 \mathbf{j} + (t+1) \mathbf{k} \), each component \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) is integrated separately:

• For the \( \mathbf{i} \) component: Integrate \( t^3 \) over the interval \([0,1]\).
• For the \( \mathbf{j} \) component: Integrate the constant \(7\) over \([0,1]\).
• For the \( \mathbf{k} \) component: Integrate \(t+1\) over \([0,1]\).

These separate integrations are then combined to form a final vector result:
\( \int_{0}^{1}(t^3 \mathbf{i} + 7 \mathbf{j} + (t+1) \mathbf{k}) \, dt = \frac{1}{4} \mathbf{i} + 7 \mathbf{j} + \frac{3}{2} \mathbf{k} \).This approach enables more straightforward calculation and understanding of the integral of vector functions.

The power rule is a fundamental integration technique that simplifies finding antiderivatives. It applies to expressions with powers of a variable. For any function \( t^n \), where \( n \) is a constant, the integration is performed as follows:
The rule states:\[ \int t^n \, dt = \frac{t^{n+1}}{n+1} + C \]where \( C \) is the constant of integration.
In the given exercise, the integration of \( t^3 \) illustrates the use of the power rule:

• \( n = 3 \), so \( \int_{0}^{1} t^3 \, dt = \frac{t^4}{4} \)
• Evaluate from 0 to 1: \( \frac{1^4}{4} - \frac{0^4}{4} = \frac{1}{4} \)

This procedure highlights how the power rule applies to simplify vector component integration effectively. Remember:

• Increase the power by one (\( n+1 \)).
• Divide by the new power.
• Apply limits for definite integrals.

Using this rule can help break down complex integrations into simpler, understandable steps.