91Ó°ÊÓ

Integrals are a fundamental concept in calculus, important for finding areas under curves, among other applications. When working with integrals, we are essentially finding the "accumulated sum" of a function.
In our exercise, the notation \( \int_{a}^{b} f(t) \, dt \) indicates we need to evaluate the integral of the function \( f(t) \) from the lower limit \( a \) to the upper limit \( b \). Each part of our vector integral has separate components, which need individual attention.
The integral of a vector-valued function can be computed by integrating each component. This is a crucial step because it simplifies the process to a sequence of familiar one-dimensional integrals. It's important to recognize and separate each part of the vector when approaching such problems.

The exponential function, denoted \( e^t \), is a special and widely used mathematical function where the base \( e \) (approximately 2.718) is raised to the power of \( t \).
It's pivotal in various fields such as growth calculations, compound interest, and in our case, integral calculus.

• Decay and Growth: Depending on the sign of the exponent, the function can model exponential growth (positive exponent) or decay (negative exponent).
• Differentiation and Integration: The derivative and the integral of \( e^t \) are unique as they preserve the function shape. For \( e^{-t} \), integrating results in \( -e^{-t} \).

In the original problem, the exponential function \( e^{-t} \) appears. To solve the integral, we utilize properties like integration of \( e^t \), adjusted for decay.

A constant term in calculus is a term that does not depend on a variable. When integrating a constant, you simply multiply it by the difference in the integration limits.
In our exercise, we have the expression \( e' \mathbf{i} \). Since \( e' \) is not a common expression, interpret it as a constant. This simplifies integration, as it remains unchanged except for being scaled by the interval size.
The result is simple to obtain: multiply the constant by the length of the interval (which, in our problem, spans from 0 to 1, equaling 1).
This reinforces the ease by which constants can be handled in an integral—they scale according to the size of the integration limits without further alteration.

A definite integral refers to the evaluation of the integral with known upper and lower bounds. The result is a number representing the "net area" between the function and the x-axis over that interval.
For vector calculus like in this problem, a definite integral gives a vector result where each component is evaluated separately over the specified bounds.

• Working with Limits: Perform the integration of each component, then substitute the bounds to find the net result.
• Combining Results: Synthesize the results of each independently evaluated component (as seen with \( e' \mathbf{i} \) and \( (1 - \frac{1}{e}) \mathbf{j} \)) into a final vector.

The original exercise shows how to balance the integration of constants and variable-dependent expressions for a complete solution, providing a clear picture from setup to final evaluation.