Problem 29 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 29-32, write the given iterated integral as an iterated integral with the indicated order of integration. $$ \int_{0}^{1} \int_{0}^{\sqrt{1-y^{2}}} \int_{0}^{\sqrt{1-y^{2}-z^{2}}} f(x, y, z) d x d z d y ; d z d y d x $$

Step-by-Step Solution

Answer

The new integral is \( \int_{0}^{1} \int_{0}^{\sqrt{1-x^2}} \int_{0}^{\sqrt{1-x^2-y^2}} f(x, y, z) \, dz \, dy \, dx \).

Step 1: Original Integral Setup

We have the iterated integral \( \int_{0}^{1} \int_{0}^{\sqrt{1-y^{2}}} \int_{0}^{\sqrt{1-y^{2}-z^{2}}} f(x, y, z) \, dx \, dz \, dy \) with the integration order of \(dx \, dz \, dy\). We need to change this to \( dz \, dy \, dx \).

Step 2: Identify New Integration Bounds

For the new order, we need to express bounds for \(z\) and \(y\) in terms of \(x\). The original bounds show dependency: \(x\) is from \(0\) to \(\sqrt{1-y^2-z^2}\), \(z\) is from \(0\) to \(\sqrt{1-y^2}\), and \(y\) from \(0\) to \(1\).

Step 3: Adjust Integration Boundaries

Analyze the volume: \(x^2 + y^2 + z^2 \leq 1\). For fixed \(x\), \(y\) and \(z\) must satisfy \(y^2 + z^2 \leq 1-x^2\). Therefore, for a given \(x\), \(z\) ranges from \(0\) to \(\sqrt{1-x^2}\), and \(y\) from \(0\) to \(\sqrt{1-x^2-z^2}\). Finally, \(x\) bounds from \(0\) to \(1\).

Step 4: Construct New Integral

The new iterated integral with the order \(dz \, dy \, dx \) is \( \int_{0}^{1} \int_{0}^{\sqrt{1-x^2}} \int_{0}^{\sqrt{1-x^2-y^2}} f(x,y,z) \, dz \, dy \, dx \). Each integration limit reflects the spatial constraint derived from earlier steps.

Key Concepts

Order of Integration

The order of integration determines how we process multiple integrals in a multivariable calculus context. In multivariable calculus, we often deal with functions of several variables, for example, three variables: \( x \), \( y \), and \( z \). When integrating such a function over a region in three-dimensional space, we must decide on the sequence of integrations. This is what is known as the order of integration.

For iterated integrals like \( \int \int \int \), the order in which you integrate matters, affecting the limits of integration and the ease of calculation. Changing the order requires changing the bounds of integration for the affected variables, making sure they reflect the same region in space.

Choosing an appropriate order sometimes simplifies the evaluation of the integral, especially if the function or limits are complicated. It's crucial to understand that while the order can be changed, compensation by altering integration bounds appropriately is necessary to ensure the same spatial region is being measured.

Integration Bounds

Integration bounds are limits that define the range over which you're integrating for each variable in an iterated integral. They are essential in multivariable calculus because they define the region of integration in space. In iterated integrals, when you switch the order of integration, you must adjust these bounds to maintain the same region of integration.

For example, if we start with bounds like \( x \) from \( 0 \) to \( \sqrt{1-y^2-z^2} \), \( z \) from \( 0 \) to \( \sqrt{1-y^2} \), and \( y \) from \( 0 \) to \( 1 \), changing the order involves reorganizing these bounds to reflect the same 3D volume in space.

The new bounds reflect relationships between variables based on the geometry of the problem. Each time you change the integration order, you need to solve for the new limits in terms of the new variable ordering, ensuring you don't change the region you are integrating over.

Multivariable Calculus

Multivariable calculus extends calculus concepts to functions of several variables. It is crucial for analyzing systems where more than one variable affects the outcome.

When dealing with multivariable functions, integrals help us calculate quantities such as volume, mass, or other physical properties that extend into two, three dimensions, or more.

Multivariable calculus uses concepts like partial derivatives, gradients, and multiple integrals to investigate how functions change, how they are related, and how they interact across dimensions.

In our case, we practice iterated integrals that help us evaluate integrals over regions defined by more complex shapes or constraints. Understanding how to work with these integrals is key to analyzing and solving practical problems in physics, engineering, economics, and other fields.

Volume Analysis

Volume analysis using iterated integrals allows us to compute the volume under a surface or within a region in three dimensions. By setting up an iterated integral, we can capture the volume of complex shapes by integrating a function over a specified region.

This involves determining the integration bounds and the order of integration correctly. In the context of our iterated integral, the region defined by \( x^2 + y^2 + z^2 \leq 1\) describes a portion of the unit sphere.

Such integrals are advantageous in computing exact values for volumes that are challenging to calculate geometrically. By iterating through variables sequentially and considering their bounds, we effectively "slice" the volume into manageable pieces, summing them back together to find the total volume. Understanding and applying volume analysis through iterated integrals is immensely beneficial in physics and engineering for solving spatial problems efficiently.