91Ó°ÊÓ

A double integral allows us to integrate over a two-dimensional area. In this exercise, we use a double integral to find the average value of a function over a rectangular region. A double integral is written as \( \int_{a}^{b} \int_{c}^{d} f(x, y) \, dy\, dx \) where \( f(x, y) \) is the function to integrate, \( [a,b] \) is the range for \( x \), and \( [c,d] \) is the range for \( y \).

For the function \( f(x, y) = \sinh x + \sinh y \) given in the exercise, we integrate first with respect to \( y \) from 0 to 2, then with respect to \( x \) from 0 to 1.

• Integrate \( \sinh x + \sinh y \) with respect to \( y \) over the interval \([0,2]\). This gives the antiderivative \( 2\sinh x + \cosh y \).
• Then, substitute the bounds 2 and 0 for \( y \), which gives us \( 2\sinh x + \cosh 2 - \cosh 0 \).
• Next, the expression \( 2\sinh x + (\cosh 2 - 1) \) is integrated with respect to \( x \) from 0 to 1.

Double integrals help us calculate quantities such as area, volume, or in this case, the average value of a function over a specified region.

Hyperbolic functions like \( \sinh x \) and \( \cosh x \) play a role similar to trigonometric functions but are defined using exponential functions.

• The hyperbolic sine function is \( \sinh x = \frac{e^x - e^{-x}}{2} \).
• The hyperbolic cosine function is \( \cosh x = \frac{e^x + e^{-x}}{2} \).

In the exercise, our function \( f(x, y) = \sinh x + \sinh y \) uses hyperbolic functions, and we need to evaluate their integral over the given region. Hyperbolic functions can be differentiated and integrated using rules similar to those for trigonometric functions.

When integrating hyperbolic functions:

• The integral of \( \sinh x \) is \( \cosh x + C \) where \( C \) is the constant of integration.
• Similarly, the integral of \( \cosh x \) results in \( \sinh x + C \).

These functions often appear in mathematics involving hyperbolic geometry or in scenarios dealing with exponential growth, enhancing their practical applications.

Rectangular region integration involves evaluating a double integral over a specific rectangle defined by its boundaries on the \( x \)- and \( y \)-axes. In this context, the region \( R = [0, 1] \times [0, 2] \) is the rectangle over which the function is averaged.

Understanding the concept of integration over rectangular regions gives you a valuable tool for summarizing how a function behaves over that region.

• The first step is to identify the limits of integration for each variable; \( x \) ranges from 0 to 1, and \( y \) ranges from 0 to 2.
• Set up the double integral by nesting one integral inside the other, choosing which variable to integrate first based on simplicity or order preferences.
• Calculating the integral involves solving the inner integral first and then proceeding to the outer integral.

Once the double integral is calculated, divide by the area of the rectangle, in this case, \( 2 \), to find the average value of the function over the region.