Problem 2 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Sketch the domains over which the integrals are defined. a \(\quad \int_{1}^{5} \int_{x}^{x^{2}} F(x, y) d y d x \quad \mathrm{~b} \quad \int_{1}^{5} \int_{y}^{y^{2}} F(x, y) d x d y \quad\) c \(\quad \int_{1}^{5} \int_{3}^{4} F(x, y) d y d x\) d \(\int_{0}^{\pi} \int_{-\sin x}^{\sin x} F(x, y) d y d x\) e \(\int_{\pi}^{2 \pi} \int_{\sin y}^{-\sin y} F(x, y) d x d y\) \(\int_{0}^{1} \int_{0}^{1-x^{2}} F(x, y) d y d x\)

Step-by-Step Solution

Answer

Sketch regions using described boundaries for each integral.

Step 1: Analyze Integral 1a

For the integral \( \int_{1}^{5} \int_{x}^{x^{2}} F(x, y) \, dy \, dx \), determine the region of integration. Here, \( x \) ranges from 1 to 5, and for each fixed \( x \), \( y \) ranges from \( x \) to \( x^2 \). This describes a region in the xy-plane bounded by the lines \( y = x \) and \( y = x^2 \) from \( x = 1 \) to \( x = 5 \).

Step 2: Analyze Integral 1b

For the integral \( \int_{1}^{5} \int_{y}^{y^{2}} F(x, y) \, dx \, dy \), \( y \) ranges from 1 to 5, and for each fixed \( y \), \( x \) ranges from \( y \) to \( y^2 \). This region is similar to 1a but flipped with respect to the roles of \( x \) and \( y \). It’s the area bounded by the curves \( x = y \) and \( x = y^2 \) from \( y = 1 \) to \( y = 5 \).

Step 3: Analyze Integral 1c

For the integral \( \int_{1}^{5} \int_{3}^{4} F(x, y) \, dy \, dx \), \( x \) ranges from 1 to 5, and for each fixed \( x \), \( y \) ranges from 3 to 4. This is a simple rectangular region in the xy-plane where the corners are at coordinates (1,3), (5,3), (1,4), and (5,4).

Step 4: Analyze Integral 1d

The integral \( \int_{0}^{\pi} \int_{-\sin x}^{\sin x} F(x, y) \, dy \, dx \) has \( x \) ranging from 0 to \( \pi \), and \( y \) for each fixed \( x \) ranges from \( -\sin x \) to \( \sin x \). The region is enclosed by the curves \( y = \pm \sin x \) and \( x = 0 \) to \( x = \pi \), forming a symmetrical region about the x-axis.

Step 5: Analyze Integral 1e

For \( \int_{\pi}^{2\pi} \int_{\sin y}^{-\sin y} F(x, y) \, dx \, dy \), \( y \) ranges from \( \pi \) to \( 2\pi \), and \( x \) ranges from \( \sin y \) to \( -\sin y \). Notice that \( x \)'s limits seem reversed; typically this means the area could be negative or indicate a change in order of limits. Graphing helps clarify that the curves \( x = \sin y \) and \( x = -\sin y \) delimit the region.

Step 6: Analyze Last Integral

For \( \int_{0}^{1} \int_{0}^{1-x^{2}} F(x, y) \, dy \, dx \), \( x \) ranges from 0 to 1, and \( y \) for each \( x \) ranges from 0 to \( 1-x^2 \). This region is under the parabola \( y = 1-x^2 \), forming a bounded shape that starts at (0,1) and is symmetrical about the y-axis.

Key Concepts

Region of Integration

In double integrals, determining the region of integration is crucial for setting up and solving the integral. This region refers to the particular area or domain in the coordinate plane over which the function is being integrated.
In the context of the given exercise, each integral is defined over a specific region, bounded by different curves or limits. For instance, in integral 1a, the region is defined by the bounds of \(x\) from 1 to 5 and \(y\) from \(x\) to \(x^2\). This creates a region enclosed between the lines \(y = x\) and \(y = x^2\), as \(x\) varies, providing a clear visual domain over which the integration takes place.

• Integral bounds determine the shape and size of the integration region.
• The order of integration (e.g., \(dy\ dx\) vs. \(dx\ dy\)) can impact how the region is described and understood.

Understanding the region of integration helps visualize the problem and ensure calculations are accurate.

Bounded Region

A bounded region is an area enclosed within certain limits or boundaries in the coordinate plane. These boundaries can be curves, lines, or other shapes defined by mathematical equations.
For example, in integral 1b, the domain is bounded by \(x = y\) and \(x = y^2\) from \(y = 1\) to \(y = 5\). This represents a region flipped in relation to its counterpart in integral 1a, due to the altered integration order.

• Bounded regions are critical in determining how an integral is set up in practice.
• These regions can be simple rectangles, as in 1c, or more complex areas described by curves in 1d and beyond.
• The concept of bounded regions is key to understanding the limitations we place on the direction and extent of integration.

Mastering the notion of bounded regions aids in solving double integrals effectively by identifying precise limits of integration.

Coordinate System

Double integrals rely on a coordinate system to map out regions on a plane, usually the Cartesian coordinate system composed of x and y axes. Understanding this system is crucial when examining how the limits and regions of integration are defined.
Each integral in the exercise uses the xy-coordinate system to set boundaries and create integration paths. The variables \(x\) and \(y\) directly describe positions and shapes, such as in integral 1d where the bounds go from \(-\sin x\) to \(\sin x\) along the y-axis.

• The coordinate system helps locate and visualize the region of integration.
• It simplifies the process of understanding exactly where an integral operates and the shape it encompasses.
• Switching between different systems, like polar coordinates, is sometimes necessary for more complex regions.

Grasping the concept of a coordinate system enhances the ability to handle integrals by grounding the solutions in a spatial context.

Curves and Surfaces

Curves and surfaces define the boundaries and upper or lower limits of integration in a double integral. These curves can be linear, like straight lines, or nonlinear, such as parabolas or sine waves.
In the context of the exercise, \(y = x\) and \(y = x^2\) describe the curves bounding the region in integral 1a. Meanwhile, \(x = \pm \sin y\) illustrates the reversed scenario for integration 1e.

• Identifying the curves and surfaces helps depict the bounded region accurately.
• Curves provide flexibility in defining complex or intricate regions of integration.
• Understanding their shape and orientation is key to setting up the correct integrals.

By comprehending how curves and surfaces interact within a coordinate system, solving integrals becomes a systematic approach to navigating between lines and shapes that define the region of interest.