Question

Write the given iterated integral as an iterated integral with the order of integration interchanged. Hint: Begin by sketching a region \(S\) and representing it in two ways, as in Example \(5 .\) $$ \int_{1 / 2}^{1} \int_{x^{3}}^{x} f(x, y) d y d x $$

Short answer

The new iterated integral is \( \int_{1/8}^{1} \int_{y^{1/3}}^{y} f(x, y) \, dx \, dy \).

Step-by-step solution

Sketch the Region S

The given iterated integral is \( \int_{1/2}^{1} \int_{x^3}^{x} f(x, y) \, dy \, dx \). Start by understanding the limits of integration. For a fixed \( x \), \( y \) ranges from \( y = x^3 \) to \( y = x \). On the x-axis, x ranges from \( 1/2 \) to \( 1 \). This describes a region in the xy-plane.

Draw Intercepts and Curves

The curves \( y = x^3 \) and \( y = x \) intersect at points where \( x^3 = x \), which gives \( x = 0 \) and \( x = 1 \). For our range from \( x=1/2 \) to \( x=1 \), these curves intersect only at \( x=1 \). So the region overlaps between these curves, with x ranging from \( 1/2 \) to \( 1 \).

Determine Limits for New Order of Integration

To switch the order of integration, fix \( y \) first. Identify the minimum and maximum values of \( y \) over the region. Since \( y = x^3 \) starts at \( x = 1/2 \) as \( (1/2)^3 = 1/8 \) and ends at \( y = x \) for \( x=1 \), the range of \( y \) is from \( 1/8 \) to \( 1 \).

Find x Limits as a Function of y

For a fixed \( y \), \( x \) ranges between the curves \( x = y^{1/3} \) to \( x = y \) for \( y \) between \( 1/8 \) to \( 1 \). This is because for a given y-value, \( x \) starts from the curve \( x = y^{1/3} \) and goes up to the line \( x = y \).

Write the New Iterated Integral

Combine the above results to write the new order of integration. The integral becomes \( \int_{1/8}^{1} \int_{y^{1/3}}^{y} f(x, y) \, dx \, dy \). This represents the integral with the order of integration interchanged.

Key concepts

Order of Integration

The order of integration refers to the sequence in which multiple integrals are evaluated in an iterated integral. In a two-variable case like the one given, the order of integration is the order in which you integrate with respect to each variable. For example, if you start integrating with respect to \(y\), followed by \(x\), this is one order of integration. Swapping this to integrate first with respect to \(x\) and then with \(y\) is another order. Changing the order of integration can sometimes simplify the calculation, allowing you to break the integral into manageable parts across a defined region. It requires re-evaluating the limits and understanding which variable depends on the other in a given region. This is a common technique when dealing with complex integrals involving multiple variables.

Limits of Integration

Limits of integration define the boundaries over which integration is performed. They are crucial in determining the area or volume that you're focusing on for your integral. In the context of an iterated integral, the limits tell you how the variables relate to each other within the region of integration. For the given problem: - The original integral has \(y\) limits from \(x^3\) to \(x\) for each fixed \(x\) from \(\frac{1}{2}\) to \(1\). The limits define a specific region in the xy-plane.When switching the order of integration, the limits change to accommodate - \(y\) ranging from \(\frac{1}{8}\) to \(1\) and- \(x\) ranging from \(y^{1/3}\) to \(y\). This process ensures that integration is performed over the same region, maintaining the original problem's integrity.

Sketching Regions

Sketching the region of integration is a helpful visual tool that aids in understanding how the variables and their limits relate. By graphing the curves you can determine the boundaries and overlapping areas in the graph, identifying the exact region over which the integration takes place. In the given problem, sketch the curves \(y = x^3\) and \(y = x\). Notice how these curves set up boundaries for \(y\) when \(x\) is fixed, and vice versa.Using sketches helps determine intersection points and visually verify limits of integration. In this case, mapping \(x\) from \(\frac{1}{2}\) to \(1\) along these functions shows the region and helps plan for the new limits after altering the order of integration. A clear sketch provides a reliable reference when solving complex integrals.

Integration Techniques

Switching the order of integration is a powerful technique when dealing with iterated integrals. It's advantageous because it can simplify the integration or work around mathematical challenges like singularities or difficult expressions. In the exercise, changing the order involves analyzing the region, determining new \(x\) and \(y\) boundaries, and applying integration accordingly. Other common techniques may involve substituting variables or using symmetry in the given function or region—which sometimes reveals shortcuts or simplifications. Thoroughly understanding and applying these strategies helps solve iterated integrals efficiently and accurately, providing deeper insights into the relationships described by the integral. These techniques are applicable not just in mathematics, but in various fields such as physics and engineering, where multi-variable integrations often represent real-world phenomena.