Problem 42 - Thomas Calculus

Question

Sketch the region of integration and write an equivalent double integral with the order of integration reversed. \begin{equation} \int_{0}^{2} \int_{-\sqrt{4-x^{2}}}^{\sqrt{4-x^{2}}} 6 x d y d x \end{equation}

Step-by-Step Solution

Answer

Reverse the limits to \(\int_{-2}^{2} \int_{0}^{\sqrt{4-y^2}} 6x \ dx \ dy\).

Step 1: Understand the Given Integral

The integral is given as \(\int_{0}^{2} \int_{-\sqrt{4-x^{2}}}^{\sqrt{4-x^{2}}} 6x \ dy \ dx\). This means that for each \(x\) in the interval \([0, 2]\), \(y\) ranges from \(-\sqrt{4-x^{2}}\) to \(\sqrt{4-x^{2}}\), which describes a semi-circle of radius 2 centered at the origin.

Step 2: Identify the Region of Integration

The region of integration is a semicircle centered at the origin with radius 2, extending from \(x=0\) to \(x=2\) and symmetric about the x-axis. To visualize this, note that \(y = \pm\sqrt{4-x^2}\) describes the upper and lower halves of a circle with equation \(x^2 + y^2 = 4\). Since we only consider \(x\) from 0 to 2, the region is the right half of this circle.

Step 3: Set Up the Reversed Integral

To reverse the order of integration, we will now choose limits first for \(x\), and then set \(y\) as the limits. For each \(y\) in the interval \([-2, 2]\), \(x\) ranges from 0 to \(\sqrt{4-y^2}\). This is because the equation \(x^2 + y^2 = 4\) rearranges to \(x = \sqrt{4 - y^2}\). The new integral is \(\int_{-2}^{2} \int_{0}^{\sqrt{4-y^2}} 6x \ dx \ dy\).

Step 4: Sketch the Region of Integration

To sketch the region, draw a circle with radius 2 centered at the origin. Shade the right half of the circle, from the positive x-axis to x = 2. This is the region described by the original integral.

Key Concepts

Region of Integration

In double integrals, the region of integration is crucial for determining where to evaluate the function.
For the given problem, the region corresponds to the area under the graphed function between specified limits.
The integral \( \int_{0}^{2} \int_{-\sqrt{4-x^{2}}}^{\sqrt{4-x^{2}}} 6x \, dy \, dx \)
describes a semi-circle on the xy-plane:

• Centered at the origin.
• With a radius of 2.
• The region extends from \(x = 0\) to \(x = 2\).

The integration region forms the right half of this circle. Understanding this part is essential for solving double integrals correctly.

Order of Integration

When evaluating double integrals, the order of integration refers to the sequence in which integration is performed.
In the current exercise, the integral is initially solved with respect to \(y\) first, and then \(x\).
This is expressed as:

• Inner integral \(d y\) goes from \(-\sqrt{4-x^{2}}\) to \(\sqrt{4-x^{2}}\).
• Outer integral \(d x\) spans from 0 to 2.

This order suits scenarios where one dimension is best handled first, based on geometric or algebraic convenience.
Being comfortable with changing the order can simplify solving more complex regions of integration.

Reversing Limits

Reversing limits is often a valuable technique in calculus when solving double integrals.
It involves flipping the order of integration to potentially simplify the computation process.
For the semicircle problem, originally integrating \(y\) before \(x\):

• The reversal involves first integrating with respect to \(x\), then \(y\).
• New bounds change to \(x\) ranging from 0 to \(\sqrt{4-y^2}\).
• Then, \(y\) covers between \(-2\) and \(2\).

The integral now reads as \(\int_{-2}^{2} \int_{0}^{\sqrt{4-y^2}} 6x \, dx \, dy\).
Reversing limits aligns with different geometric or algebraic aspects and can thus streamline the integral evaluation.

Semicircle

In the exercise, understanding the shape of a semicircle is pivotal.
Here, a semicircle refers to half of a full circle, precisely cut along its diameter.
The full circle is defined by the equation \(x^2 + y^2 = 4\):

• The radius is 2, meaning any point within this boundary satisfies the circle's equation.
• The semicircle described in our problem lies on the right side because \(x\) spans from 0 to 2.
• The curve is symmetric about the x-axis, ensuring that for any \(x\), two \(y\) values correspond.

Grasping semicircles and their properties assists in correctly sketching the integration region and setting appropriate limits.

Mathematical Visualization

Mathematical visualization is an important skill for tackling double integrals, allowing the interpretation of shapes and areas from algebraic expressions.
Visualizing the integration region not only provides clarity but also often reconciles complicated abstract problems.
For such problems:

• Drawing the xy-plane helps in identifying boundary conditions.
• The semicircle can be shaded, facilitating the understanding of the limits within this region.
• Visual tools make it easier to change the order of integration.

Articulating these regions visually bridges the gap between theory and practical implementation, paving the way for solving integrals with confidence.