91Ó°ÊÓ

When working with vector functions, like the given vector \( \mathbf{r} = 3 \sin t \mathbf{i} - \cos t \mathbf{j} + (2-t) \mathbf{k} \), vector integration involves integrating each of the components of the vector separately. This process is essential in vector calculus whenever you need to find the accumulated effect of a vector quantity over a particular range. One of the primary steps in vector integration is identifying the component functions. For \( \mathbf{r} \), these functions are:

• \( 3 \sin t \) for the \( \mathbf{i} \) component
• \( -\cos t \) for the \( \mathbf{j} \) component
• \( 2-t \) for the \( \mathbf{k} \) component

By integrating these scalar functions separately, we can obtain the integral of the entire vector function over the specified interval. It's crucial to keep the operations on each component separated until the end, where results are combined into a single vector. These integrations can provide insight into various physical phenomena, such as calculating the net force, displacement, or flux across a surface in a defined region.

Component-wise integration is essentially a method in vector calculus that simplifies the integration process by breaking down a vector into its individual components. Each component, represented often as a scalar function, is treated individually. Let's understand how it works using the vector function \( \mathbf{r}(t) \). The integration process occurs as follows:

• Identify the Components: - \( \mathbf{i} \) component: \( 3 \sin t \) - \( \mathbf{j} \) component: \( -\cos t \) - \( \mathbf{k} \) component: \( 2-t \)
• Integrate Each Component Separately: - Integrate \( 3 \sin t \) over the interval: Results in \( 6 \mathbf{i} \) - Integrate \( -\cos t \) over the interval: Results in \( 0 \mathbf{j} \) - Integrate \( 2-t \) over the interval: Results in \( (2\pi - \frac{\pi^2}{2}) \mathbf{k} \)
• Combine Results: Add the results of the component integrals to address the vector as a whole.

This process ensures that each part of the vector is accurately accounted for, and the true value of the integral over the specified interval is obtained.

The definite integral is a fundamental concept in calculus, crucial for calculating the precise accumulation of quantities. In the context of vector calculus, definite integrals are used to determine the total effect of a vector function over a defined interval. In our example, the definite integral \( \int_{0}^{\pi} \mathbf{r} \mathrm{d}t \) represents the accumulation of the vector function \( \mathbf{r}(t) \) from \( t=0 \) to \( t=\pi \). Here are some essential points regarding definite integrals:

• Bounded Limits: Definite integrals compute the net change from one limit to another, simplifying to a numerical value or a combination of values across components.
• Evaluation at Bounds: After integrating a component, you evaluate at the upper limit and subtract the evaluation at the lower limit, as shown: \[\int_{0}^{\pi} 3 \sin t \, \mathrm{d}t = 3[-\cos t]\bigg|_0^{\pi} = 6.\]
• Physical Interpretations: - In physics, definite integrals can represent displacement, work done, or total charge, acting as a significant component in practical applications.

This foundational aspect of calculus allows us to derive comprehensive results in engineering, physics, and various fields requiring precise calculations.