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 \(5 .\) $$ \int_{0}^{1} \int_{x^{2}}^{x^{1 / 4}} f(x, y) d y d x $$

Short answer

The iterated integral is \( \int_{0}^{1} \int_{y^4}^{\sqrt{y}} f(x, y) \, dx \, dy \).

Step-by-step solution

Identify the Region of Integration

We start by identifying the limits of integration in the given iterated integral \( \int_{0}^{1} \int_{x^{2}}^{x^{1/4}} f(x, y) \, dy \, dx \). The region \( S \) in the \( (x, y) \)-plane is defined by the inequalities: \( 0 \leq x \leq 1 \) and \( x^2 \leq y \leq x^{1/4} \).

Sketch the Region

Next, we sketch the region \( S \) in the \( (x, y) \)-plane. The curve \( y = x^2 \) is a parabola opening upwards, while \( y = x^{1/4} \) is a curve that rises more slowly than the linear function and is above the parabola for \( 0 < x < 1 \). The region \( S \) is between these two curves from \( x = 0 \) to \( x = 1 \).

Re-Express the Region for Alternate Integration

When reversing the order of integration, we need to express \( x \) as a function of \( y \). From the equations \( y = x^2 \) and \( y = x^{1/4} \), solve for \( x \) in terms of \( y \): \( x = \sqrt{y} \) and \( x = y^4 \). The region is bounded by these curves from \( y = 0 \) to \( y = 1 \).

Set Up the New Iterated Integral

The new order of integration is \( dy \, dx \). For a fixed \( y \), \( x \) ranges from \( y^4 \) to \( \sqrt{y} \), as these functions describe the left and right boundaries of region \( S \), respectively. Thus, the new integral is \( \int_{0}^{1} \int_{y^4}^{\sqrt{y}} f(x, y) \, dx \, dy \).

Write the Final Answer

The iterated integral with the order of integration interchanged is given by:\[\int_{0}^{1} \int_{y^4}^{\sqrt{y}} f(x, y) \, dx \, dy\]

Key concepts

Iterated Integral

An iterated integral is a method used in calculus to evaluate integrals with more than one variable. It's essentially performing multiple integrations in a sequence. When dealing with iterated integrals, you'll often encounter expressions of the form \(\int \int f(x, y)\, dy\, dx\), which tell you to integrate first with respect to \(y\) while treating \(x\) as a constant, and then integrate the result with respect to \(x\).
The process:

• Perform the innermost integral first.
• Apply the limits of integration for that variable.
• Continue with the next variable, using the result of the previous integration.
• Finally, apply the outer limits.

Iterated integrals can also involve more than two variables, allowing integration over multidimensional spaces, simplifying complex problems in engineering and physics.

Order of Integration

The order of integration in an iterated integral determines which variable you integrate first. The exercise asked us to start with \(dy\, dx\), meaning we first integrate with respect to \(y\) while keeping \(x\) constant. Then, we integrate the result with respect to \(x\).
Changing the order of integration:

• Involves analyzing the region of integration.
• Requires re-expressing limits based on the alternate variable sequence.

This process is crucial in some situations, as it can simplify calculations or make an integral easier to evaluate. Sometimes, the integral may not be solvable in one order but may become accessible when reversed.

Integration Boundaries

The integration boundaries define the limits over which the integration occurs. In the original problem, these boundaries are given as functions of \(x\) and \(y\): between \(x=0\) and \(x=1\), and between the curves \(y=x^2\) and \(y=x^{1/4}\).
Why integration boundaries matter:

• They describe the "slice" of the plane over which we integrate.
• Correctly defining them ensures the integral evaluates the desired region properly.

When reversing the order, these boundaries must also be re-evaluated in terms of the new variable order, ensuring consistency with the region of integration.

Region of Integration

Understanding the region of integration is key to properly setting up and switching iterated integrals. In the given exercise, the region \(S\) is bounded by \(y=x^2\) (a parabola) and \(y=x^{1/4}\) (a slowly rising curve). It spans from \((0,0)\) to \((1,1)\).
Steps to define the region:

• Sketch the curves to visualize where they intersect and how they bound the region.
• Identify the domain for each variable across the overlapping curves.
• Examine the limits of integration, dependent on the chosen variable for integration.

In reversing the integration order, keep the confines of the region intact by skillfully re-evaluating boundary limits for the new order: in our case, by determining \(x\) values in terms of \(y\). This guarantees the integral still represents the same geometric area.