Problem 16 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the solid S. Then write an iterated integral for $$ \iiint_{S} f(x, y, z) d V $$ $$ S=\left\\{(x, y, z): 0 \leq x \leq y^{2}, 0 \leq y \leq \sqrt{z}, 0 \leq z \leq 1\right\\} $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Solid's Boundaries

First, let's identify the boundaries of the solid S based on the provided inequalities. We have three conditions: \(0 \leq x \leq y^2\), \(0 \leq y \leq \sqrt{z}\), and \(0 \leq z \leq 1\). This implies that for a fixed \(z\), \(y\) ranges from 0 to \(\sqrt{z}\), and for each \(y\), \(x\) ranges from 0 to \(y^2\). The boundary for \(z\) is between 0 and 1.

Step 2: Sketch the Solid S

The conditions \(0 \leq x \leq y^2\) and \(0 \leq y \leq \sqrt{z}\) define a region in 3D space. \(x = y^2\) is a parabolic surface, \(y = \sqrt{z}\) is a semicircular cap in the positive yz-plane, and \(z = 1\) is the top cap of the solid. The solid lies inside these surfaces, creating a shape that stretches up to \(z = 1\) and has a parabolic base in the xy-plane.

Step 3: Write the Iterated Integral

To write the iterated integral for \(\iiint_{S} f(x, y, z) \, dV\), integrate the function first with respect to \(x\), then \(y\), and finally \(z\). The iterated integral is given by: \[ \int_{0}^{1} \int_{0}^{\sqrt{z}} \int_{0}^{y^2} f(x, y, z) \, dx \, dy \, dz \].This order ensures the integration respects the bounds for each variable as identified in the problem.

Key Concepts

Triple Integrals

Triple integrals extend the concept of integration to functions of three variables over a three-dimensional region. Think of it as calculating the "volume under a surface," but in three dimensions rather than two. When you tackle a problem involving triple integrals, you typically begin by specifying the order of integration, which depends on the function and the region of integration. You might integrate with respect to one variable first, while holding the others constant, and then proceed with the remaining variables. This problem illustrates integrating a function with variables \(x, y, z\) over a complex region \(S\).

The given iterated integral \[ \int_{0}^{1} \int_{0}^{\sqrt{z}} \int_{0}^{y^2} f(x, y, z) \, dx \, dy \, dz \] suggests that for each fixed \(z\), you first integrate over \(x\) from \(0\) to \(y^2\). Next, you integrate over \(y\), ranging from \(0\) to \(\sqrt{z}\), and finally over \(z\) from \(0\) to \(1\). This order is chosen based on the geometric constraints of the solid \(S\). It's crucial to approach the integral in this order to respect the natural boundaries of the solid.

Solid Boundaries

One of the most important parts of evaluating triple integrals is understanding the boundaries of the solid. The boundaries define the limits within which each variable may vary, and they are crucial for setting up the correct iterated integral. In this problem, the region \(S\) is defined by the inequalities:

• \(0 \leq x \leq y^2\)
• \(0 \leq y \leq \sqrt{z}\)
• \(0 \leq z \leq 1\)

Each inequality describes a slice or segment of the solid:

• The inequality \(0 \leq x \leq y^2\) represents that \(x\) is limited by \(y^2\), forming a parabolic surface.
• The inequality \(0 \leq y \leq \sqrt{z}\) implies that \(y\) changes from \(0\) to \(\sqrt{z}\), creating a semi-circle on the yz-plane.
• The inequality \(0 \leq z \leq 1\) indicates that \(z\) ranges vertically from \(0\) to \(1\), forming the height of the solid.

Boundary Interaction

The interaction of these boundaries determines the overall shape of the solid. It’s helpful to visualize the intersection of the surfaces that each inequality defines, as this aids in correctly setting up the integration limits.

3D Sketching

Sketching a three-dimensional region such as this one allows you to better understand the solid and its boundaries. For this problem, you start by picturing some of the familiar geometric shapes mentioned:

• \(x = y^2\): a parabolic surface in the xy-plane.
• \(y = \sqrt{z}\): a semicircular shape in the yz-plane.
• \(z = 1\): the horizontal plane at the top of the solid.

In essence, the solid \(S\) is bounded by these surfaces and exists in the region of space where these inequalities overlap. The plane \(z = 1\) acts as a ceiling, while the other boundaries form the sides and base of the solid below this height. Each of these surfaces gradually helps form a mental picture of the solid's dimensions and how it fits in space.

Visualizing the Shape

If you have trouble visualizing this, try sketching each individual surface first, then determine where they intersect. Start by drawing the parabolic curve \(x = y^2\) on a 2D plane, then introduce the boundary \(y = \sqrt{z}\) to see how it limits \(y\). Finally, extend your visualization into three dimensions by including the flat plane \(z = 1\). This layered sketching approach helps students to see how the flat and curved boundaries culminate into the final solid.

Calculus Concepts

Understanding calculus concepts is essential for grasping complex integrations like triple integrals. These concepts include differentiation and integration, limits, and understanding the behavior of functions over a region. In triple integrals, you apply these concepts in three dimensions, which can be challenging but rewarding.

• Differentiation and Integration: Differentiation helps understand how functions behave at any given point, while integration aggregates values over a region. Together, they allow you to find volumes or other aggregated values over 3D spaces.
• Limits: The concept of limits ensures that the function behaves consistently as it approaches specific boundaries within the integral. Proper understanding of limits also allows for correct setup of integration bounds.
• Function Behavior: Understanding how functions behave and change over different regions helps in predicting outcomes and verifying the results of the integral equations.

With these calculus concepts, one can tackle complex problems involving volumes and other physical properties, ensuring that solutions respect both mathematical laws and physical realities. This knowledge is crucial not only in solving textbook exercises but also in real-world applications where 3D modeling and calculations are necessary.