Question

Reverse the order of integration in the following integrals. $$\int_{1 / 2}^{1} \int_{0}^{-\ln y} f(x, y) d x d y$$

Short answer

Question: Reverse the order of integration for the given integral: $$\int_{1 / 2}^{1} \int_{0}^{-\ln y} f(x, y) d x d y$$ Answer: The reversed integral, changing the order of integration from dy dx to dx dy, is: $$\int_{0}^{-\ln\left( \frac{1}{2} \right)} \int_{e^{-x}}^{1} f(x, y) d y d x$$

Step-by-step solution

Analyze the given integral

First, let's identify the current limits of integration. The given integral is in dy dx form: $$\int_{1 / 2}^{1} \int_{0}^{-\ln y} f(x, y) d x d y$$ The inner integral is with respect to x, with limits 0 and \(-\ln y\). The outer integral is with respect to y, with limits \(\frac{1}{2}\) and 1.

Find the corresponding equations

For the inner integral, the limits of integration can be written as equations: $$x = 0$$ $$x = -\ln y$$ For the outer integral, the limits of integration can be written as equations: $$y = \frac{1}{2}$$ $$y = 1$$

Rearrange equations to reverse order of integration

To reverse the order of integration to dx dy, we need to express the above equations in terms of y: For \(x = -\ln y\), solving for y, we get: $$y = e^{-x}$$ Now we have the equations needed to reverse the order of integration: $$x = 0$$ $$y = e^{-x}$$ $$y = \frac{1}{2}$$ $$y = 1$$

Determine new limits of integration

Using the transformed equations, we can now find the new limits of integration for the reversed integral in dx dy form: For the inner integral (in terms of y): Lower limit: \(y = e^{-x}\) Upper limit: \(y = 1\) For the outer integral (in terms of x): Lower limit: \(x = 0\) Upper limit: \(x = -\ln \frac{1}{2}\) (since when \(y=\frac{1}{2}\), \(x = -\ln y = -\ln \frac{1}{2}\))

Write down the reversed integral

Now using the new limits of integration, we can write the reversed integral in dx dy form as: $$\int_{0}^{-\ln\left( \frac{1}{2} \right)} \int_{e^{-x}}^{1} f(x, y) d y d x$$

Key concepts

Reversing Integral Order

Reversing the order of integration can make certain double integrals easier to evaluate. It involves switching the roles of the inner and outer integrals, which ultimately requires changing the limits of integration.

This process usually happens in five steps:

• First, identify the current limits of integration for both variables, often dealing with one variable at a time.
• Second, express these limits in terms of corresponding equations.
• Third, solve these equations to express one variable in terms of the other.
• Fourth, determine the new limits of integration from these transformations.
• Finally, rewrite the integral with the new variable order and limits.

Remember, this transformation is crucial when the initial integral is challenging to solve directly due to complicated regions of integration.

Limits of Integration

Limits of integration represent the boundaries of the region over which we are integrating. In double integrals, these limits dictate which area or volume of space is being evaluated.

When reversing the order of integration, the determination of new limits is crucial so that the same region is evaluated. The previous limits will be treated as equations that can be manipulated to reflect the area properly when their order is reversed.

• The inner integral's limits are derived from the values of the variable that's solved last in the original setup.
• The outer integral limits are the initial boundaries of the outer segment of integration but adjusted to reflect the new equations developed earlier during rearrangements.

Special attention to these limits is necessary to ensure accuracy in calculating the integral.

Functions of Two Variables

Functions of two variables, denoted as \(f(x, y)\), are integral to understanding areas and volumes in multivariable calculus. Each output value of such a function relies significantly on the pair of input values of \(x\) and \(y\).

These functions are often explored within specific regions, bounded by constraints or limits of integration. Understanding how these functions behave within different parts of their domain is essential when evaluating integrals.

• Each input combination \((x, y)\) gives us a value describing a surface's height at that particular point when the function represents a surface in three-dimensional space.
• When integrated over a region, the function's output can determine quantitative properties like volume or surface area, depending on the context.
• The manipulation of integration through order reversal allows for better examination of these functions within complex or irregular regions.

Thus, double integrals provide a deeper insight into areas described by such functions by efficiently managing variable interactions.