Problem 12 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 11-20, sketch the solid \(S\). Then write an iterated integral for $$ \iiint_{S} f(x, y, z) d V $$ $$ \begin{gathered} S=\left\\{(x, y, z): 0 \leq x \leq \sqrt{4-y^{2}}\right. \\ 0 \leq y \leq 2,0 \leq z \leq 3\\} \end{gathered} $$

Step-by-Step Solution

Answer

The iterated integral is \( \int_{0}^{3} \int_{0}^{2} \int_{0}^{\sqrt{4-y^2}} f(x, y, z) \ dx \ dy \ dz \).

Step 1: Understanding the Region in the xy-plane

The inequality \(0 \leq x \leq \sqrt{4-y^2}\) describes a region in the xy-plane. Set \(x = \sqrt{4-y^2}\). When squaring both sides, we get \(x^2 + y^2 = 4\), which is a semi-circle of radius 2 centered at the origin. Since \(0 \leq y \leq 2\), only the upper half of the circle is considered.

Step 2: Interpreting the Solid Region S

The solid \(S\) is bounded by the surfaces: \(0 \leq x \leq \sqrt{4-y^2}\), \(0 \leq y \leq 2\), and \(0 \leq z \leq 3\). Thus, for each \((y,z)\), \(x\) ranges from 0 to the semi-circle \(\sqrt{4-y^2}\). \(y\) ranges from 0 to 2, and \(z\) ranges independently from 0 to 3.

Step 3: Sketch the Solid S

The region is half of a cylinder (since \(x\) and \(y\) describe the semi-circular cross-section with a height along the \(z\)-axis). Sketch a semicircle with diameter from \((-2,0)\) to \((2,0)\) in the xy-plane and then extend vertically up to \(z=3\).

Step 4: Setting up the Iterated Integral

The integral is written as \[ \iiint_{S} f(x, y, z) dV = \int_{0}^{3} \int_{0}^{2} \int_{0}^{\sqrt{4-y^2}} f(x, y, z) \, dx \, dy \, dz \]. We integrate \(x\) from 0 to \(\sqrt{4-y^2}\), \(y\) from 0 to 2, and \(z\) from 0 to 3.

Key Concepts

Iterated Integral

An iterated integral is a way to compute the volume of a region in a three-dimensional space, often involving multiple integrals arranged in a nested structure. In the context of this problem, calculating the triple integral requires evaluating the integrals sequentially over the specified bounds.
Start with the inner-most integral, which in this exercise is with respect to the variable \(x\). It ranges from 0 to \(\sqrt{4-y^2}\), representing a part of the solid \(S\). Then, move on to the \(y\)-coordinate, which ranges from 0 to 2, followed by \(z\), which spans independently from 0 to 3.
These integrals are evaluated in sequence, making them nested or iterated. Such an approach is particularly helpful for analyzing complex shapes and volumes in multivariable calculus as it breaks down challenging problems into manageable steps.

Solid of Revolution

A solid of revolution is formed when a two-dimensional shape is rotated around an axis.
In this exercise, the solid is akin to a semi-cylindrical shape derived from the inequality \(0 \leq x \leq \sqrt{4-y^2}\). This describes a semi-circle in the \(xy\)-plane with a radius of 2 centered at the origin. By extending this shape vertically along the \(z\)-axis from \(z = 0\) to \(z = 3\), a half-cylinder-like solid is produced.
Such forms are crucial for understanding volume computation techniques in calculus, as they often arise in applications involving rotational symmetry.

Cylindrical Coordinates

The cylindrical coordinate system provides a useful way to describe certain solids, especially those involving circles or cylindrical shapes in three-dimensional space.
In this coordinate system, a point in space is represented by \((r, \theta, z)\), where \(r\) is the radius from the \(z\)-axis, \(\theta\) is the angle around the \(z\)-axis, and \(z\) denotes the height. This differs from Cartesian coordinates which use \((x, y, z)\).
For the given problem, while a Cartesian system is used, thinking of the region in terms of cylindrical coordinates might simplify imagining or manipulating the geometry, since the semi-circular base finds a natural description as a radius \(r = 2\) in cylindrical coordinates.

Multivariable Calculus

Multivariable calculus extends traditional calculus to functions of several variables, providing tools for exploring diverse problems in physics, engineering, and more.
In this context, notions such as gradients, differentials, and integrals are expanded to account for multiple dimensions. This exercise exemplifies these principles through the use of nested integrals over a three-dimensional solid.
By moving through each dimension via an iterated integral, students learn how to navigate the complexities of calculating volumes in higher dimensions, stepping beyond the familiar single-variable calculus to explore richer problems.