Problem 31 - Calculus an Applied Approach

Question

Sketch the region \(R\) whose area is given by the double integral. Then change the order of integration and show that both orders yield the same area. $$ \int_{0}^{1} \int_{y^{2}}^{\sqrt[3]{y}} d x d y $$

Step-by-Step Solution

Answer

The area of the region R is 1/6 whether computed using the original order of integration or after switching the order, confirming Fubini's theorem.

Step 1: Sketching the Region R

R is defined between the curves \( x = y^{2} \) and \( x = \sqrt[3]{y} \) with y ranging from 0 to 1. On sketching these curves on the xy-plane, the region is clearly visible.

Step 2: Compute Original Double Integral

Now compute the original double integral \[\int_{0}^{1} \int_{y^{2}}^{\sqrt[3]{y}} d x d y \]This is done by integrating first w.r.t x (from \( y^{2} \) to \( \sqrt[3]{y} \)), followed by y (from 0 to 1). The answer should be 1/6.

Step 3: Change Order of Integration

Next, change the order of integration from \( dy \, dx \) to \( dx \, dy \). This results in a new set of integration limits. The integral is now \[\int_{0}^{1} \int_{x^{3}}^{\sqrt{x}} d y d x \]

Step 4: Compute Changed Double Integral

Now compute the changed double integral. Again, integrate first w.r.t y (from \( x^{3} \) to \( \sqrt{x} \)), followed by x (from 0 to 1) as before. The answer should also be 1/6.

Step 5: Compare the Results

The results from the original integral and after changing the order are the same, showing the validity of Fubini's theorem. This theorem says that in most usual conditions the order of integration does not change the result.

Key Concepts

Order of Integration

When dealing with double integrals, it's essential to understand what the order of integration means. The **order of integration** refers to the sequence in which you integrate the variables. In double integrals, you will typically see limits outlined for integrating one variable first and then the other.
In our exercise, the original order is \( dx \, dy \). This means we first integrate with respect to \( x \) from \( y^2 \) to \( \sqrt[3]{y} \), and then with respect to \( y \) from 0 to 1. However, by switching the order, we change the variable of first integration to \( y \), offering different limits of integration.
For the switched order, \( dy \, dx \), we integrate \( y \) first, over the range from \( x^3 \) to \( \sqrt{x} \), and then \( x \) from 0 to 1. This switch is crucial in scenarios where one order may be easier to solve than the other. Despite this change, the area covered, or the final result of the integral, remains the same due to the properties that govern double integrals.

Sketching Regions

Understanding the region of integration is crucial when working with double integrals. **Sketching Regions** involves drawing the curves or lines that define the boundaries within which you are integrating.
In this problem, the region \( R \) is bounded by the curves \( x = y^2 \) and \( x = \sqrt[3]{y} \). To visualize how these curves define the region, sketch them on an \( xy \)-plane. Begin by plotting a few points for each equation:

• For \( x = y^2 \), try points (0,0), (1,1), and so forth.
• For \( x = \sqrt[3]{y} \), plot points (0,0), (1,1), etc.

The area between these curves from \( y = 0 \) to \( y = 1 \) will show you the region of integration. By nature, seeing the region visually helps in understanding the limits for integration when switching the order, guiding which limits to replace or set.

Fubini's Theorem

**Fubini's Theorem** is a fundamental result in calculus that plays a pivotal role in evaluating double integrals. The theorem states that, given certain conditions where the function is continuous or on a region of finite measure, the order of integration in a double integral can be switched without affecting the outcome.
In our exercise, Fubini's Theorem guarantees that both the original integral \( \int_{0}^{1} \int_{y^{2}}^{\sqrt[3]{y}} dx \, dy \) and the switched one \( \int_{0}^{1} \int_{x^{3}}^{\sqrt{x}} dy \, dx \) yield the same area, which is \( \frac{1}{6} \). This principle is useful as sometimes one order presents a significantly simpler computation than the other.
Fubini's theorem provides assurance and flexibility. It ensures we can adapt our approach to suit simpler calculations, crucial for optimizing efforts and preventing potential errors when solving more complicated integrals.