Problem 28 - Thomas Calculus

Question

In Exercises \(21-30,\) sketch the region of integration and write an equivalent double integral with the order of integration reversed. $$ \int_{0}^{2} \int_{0}^{4-y^{2}} y d x d y $$

Step-by-Step Solution

Answer

The reversed order integral is: \( \int_{0}^{4} \int_{0}^{\sqrt{4-x}} y \, dy \, dx \).

Step 1: Identify the region of integration

The original integral is given with limits \(0 \leq y \leq 2\) and for each fixed \(y\), from \(x = 0\) to \(x = 4 - y^2\). Thus, the region describes a portion of the plane bounded by \(x = 0\), \(x = 4 - y^2\) (a downward opening parabola), \(y = 0\), and \(y = 2\).

Step 2: Sketch the region

To sketch the region, plot the parabola \(x = 4 - y^2\) by transforming it to \(y^2 = 4 - x\). The parabola opens to the left with its vertex at (4,0) and intersects the y-axis at (0,2) and (0,-2). However, since \(y\) ranges from 0 to 2, the relevant portion is the upper half of the parabola from \(x = 0\) to \(x = 4\) and \(y = 0\) to \(y = 2\).

Step 3: Determine new limits of integration

When reversing the order of integration, observe that for every \(x\), \(y\) varies from the lower boundary \(y = 0\) to the upper boundary \(y = \sqrt{4 - x}\) from the given parabola. The bounds for \(x\) are from left vertical line \(x=0\) to the right vertical line \(x=4\). Thus, \(x\) limits are from 0 to 4.

Step 4: Write the new double integral

The reversed integral, switching the order of integration from dydx to dxdy, is: \[ \int_{0}^{4} \int_{0}^{\sqrt{4-x}} y \, dy \, dx \]. This represents the same region by integrating \(y\) from 0 to \(\sqrt{4-x}\) and \(x\) from 0 to 4.

Key Concepts

Reversing Integration Order

When solving double integrals, the order of integration can sometimes be reversed to simplify the process or to fit the constraints of a problem better. To reverse the order of integration, we need to reassess the limits of both variables by observing the geometric region of integration. In the given problem, originally, we integrate over \( y \) first, then \( x \), with the bounds \( 0 \leq y \leq 2 \) and \( 0 \leq x \leq 4 - y^2 \). Thus, \( y \) is first integrated without dependence on another variable. To reverse, \( x \) should have definite bounds (independent), while \( y \) should functionally depend on \( x \). We discover that for every \( x \), \( y \) ranges from 0 to \( \sqrt{4 - x} \), describing a new region where \( x \) goes from 0 to 4 and \( y \) adjusts dynamically by \( x \). Reversing the order involves changing the structure of the problem itself:

• Analyzing how the bounds of each variable were dependent on the other initially.
• Re-sketching or mentally visualizing the region in the new coordinate structure.
• Rewriting the integral equation to reflect this swapped order.

This flexibility in changing the integration order aids immensely in solving complicated integrals efficiently.

Parabolic Regions

In this exercise, the integral refers to a region constrained by a parabolic boundary. The parabola given is originally expressed as \( x = 4 - y^2 \). This is a left-opening parabola reflecting a downward shift from the typical \( x = y^2 \) form. When translated to the more familiar form \( y^2 = 4 - x \), it becomes apparent how the parabola sits within the Cartesian plane. Parabolas like these delineate curved zones that we must navigate during integration. Understanding the geometry helps in determining accurate integration limits. Here are steps to handle parabolic regions efficiently:

• **Translate** parabolic equations to familiar forms to better understand their shapes.
• **Identify** key points like vertices and axes intersections, crucial in limiting integration bounds.
• **Visualize** the region within the geometric plane to map out sections for calculation effectively.

The critical takeaway in handling parabolic regions in integrals is applying geometry to adjust boundaries and appreciate the region's uniqueness within the plane.

Limits of Integration

Limits are crucial in integrating functions over specific regions. They define where the function is evaluated and constrain it to a particular section of the graph. For double integrals like this one, the limits will encapsulate all variables involved. For the given problem:- Initially, \( y \) ranged from 0 to 2, and for each \( y \), \( x \) was within \( 0 \) to \( 4 - y^2 \). This aligns the focus vertically, first capturing depth then breadth.- Reversing to \( x \) from 0 to 4 necessitates \( y \)'s functional range from 0 to \( \sqrt{4 - x} \), essentially working horizontally from bottom to top as \( x \) traverses. Setting limits accurately involves:

• **Analyzing the boundaries** of the original geometric region.
• **Deducting dependencies**; determine if one variable's limits are functions of another.
• **Rewriting limits** to fit the new integration order if reversed, ensuring the entire region remains covered.

Mastering how limits fit within each integral setup is key to solving double integrals correctly and efficiently. Proper limit conceptualization ensures the integrity of the integration process and accurately captures the target region.