91Ó°ÊÓ

A definite integral is a key concept in calculus. It helps us find the area under a curve over a particular interval. For example, in our exercise, we find the area under the curve of the function \( y = e^{x} + e^{-x} \) from \( x = -1 \) to \( x = 1 \). The notation for a definite integral looks like this: \[ A = \int_{-1}^{1} (e^{x} + e^{-x}) \, dx \] This integral tells us to find the sum of areas, or more specifically, the net area, between the curve and the x-axis, between the limits of integration which are -1 and 1.

Basic rules you'll need include:

• The Fundamental Theorem of Calculus, which links differentiation and integration.
• Setting up the integral based on the given function and interval.

Definite integrals can be evaluated by finding the antiderivatives (indefinite integrals) of the function, and then using the limits of integration. This process finds the accumulated area under the curve from lower to upper limits.

Exponential functions are a type of mathematical function where the variable appears as an exponent. Common exponential functions include \( e^{x} \), where \( e \) is the base of natural logarithms (approximately 2.718). In this exercise, the function is \( y = e^{x} + e^{-x} \).

Key properties of exponential functions are:

• They grow rapidly either positively or negatively, depending on whether the exponent is positive or negative.
• They are always positive for real values of x.
• They have an inverse called the logarithmic function.

The exponential function \( e^{x} \) increases quickly as \( x \) becomes larger and decreases as \( x \) becomes more negative. On the other hand, \( e^{-x} \) decreases as \( x \) increases. Combining \( e^{x} \) and \( e^{-x} \) results in a symmetric curve about the y-axis, since \( f(x) = f(-x) \). This symmetry simplifies our calculations and sketches.

Graphical region analysis involves sketching the curve given by a function and identifying regions of interest under the curve. For the function \( y = e^{x} + e^{-x} \), we focus on the interval \( -1 \leq x \leq 1 \). The following steps help in analyzing such regions:

• Draw the graph of the function within the given interval. Notice if the function is symmetric or has any particular shape which can simplify calculations.
• Shade the area under the curve between the specified interval. This highlighted region is where the area will be calculated using integration.
• Evaluate the definite integral to compute the total area of this region.

For instance, after drawing the curve for \( y = e^{x} + e^{-x} \), we observe symmetry around the y-axis. This observation tells us that analyzing just one side (either positive or negative side of the x-axis) and doubling the answer might be an effective strategy. By evaluating the integral from \( -1 \) to \( 1 \), we ensure we accurately capture the total area under the function \( y \) for the given x-range.