Question

Rewrite the following integrals using the indicated order of integration and then evaluate the resulting integral. $$\int_{0}^{1} \int_{0}^{\sqrt{1-x^{2}}} \int_{0}^{\sqrt{1-x^{2}}} d y d z d x \text { in the order } d z d y d x$$

Short answer

Question: Rewrite the order and evaluate the triple integral $$\int_{0}^{1} \int_{0}^{\sqrt{1-x^{2}}} \int_{0}^{\sqrt{1-x^{2}}} d y d z d x$$ in the order of \(dz \, dy \, dx\). Answer: $\frac{4}{15}$

Step-by-step solution

Identify integration limits of given integral

The given integral is $$\int_{0}^{1} \int_{0}^{\sqrt{1-x^{2}}} \int_{0}^{\sqrt{1-x^{2}}} d y d z d x .$$ The corresponding integration limits for \(x\), \(y\) and \(z\) are: - For \(x\), the range is \(0 \le x \le 1\) - For \(y\), the condition is \(0 \le y \le \sqrt{1-x^{2}}\) - For \(z\), the condition is \(0 \le z \le \sqrt{1-x^{2}}\)

Rewrite integration limits in the order of \(dz \, dy \, dx\)

We have the bounding conditions: $$0 \le x \le 1$$ $$0 \le y \le \sqrt{1-x^{2}}$$ $$0 \le z \le \sqrt{1-x^{2}}$$ In order to find the corresponding limits for \(dz \, dy \, dx\), we have to analyze the bounding conditions. As \(z\) is solely dependent on \(x\), the condition for z remains the same: $$0 \le z \le \sqrt{1-x^{2}}$$ Since \(y\) is also solely dependent on \(x\), the condition for y remains the same: $$0 \le y \le \sqrt{1-x^{2}}$$ And for \(x\), we only need to find the minimum and maximum values. As we can tell from the problem, the minimum value for x is \(0\) and the maximum value is \(1\): $$0 \le x \le 1$$ Now, we can rewrite the integral in the order \(dz \, dy \, dx\) as: $$\int_{0}^{1} \int_{0}^{\sqrt{1-x^{2}}} \int_{0}^{\sqrt{1-x^{2}}} d z d y d x$$

Evaluate the new integral

Evaluating our new triple integral: $$\int_{0}^{1} \int_{0}^{\sqrt{1-x^{2}}} \int_{0}^{\sqrt{1-x^{2}}} d z d y d x$$ $$= \int_{0}^{1} \int_{0}^{\sqrt{1-x^{2}}} \left[z\right] \Big|_{0}^{\sqrt{1-x^{2}}} dy\, dx $$ $$= \int_{0}^{1} \int_{0}^{\sqrt{1-x^{2}}} \left(\sqrt{1-x^{2}}\right) dy\, dx $$ $$= \int_{0}^{1} \left[\frac{1}{2}y\left(1-x^{2}\right)\right] \Big|_{0}^{\sqrt{1-x^{2}}} dx $$ $$= \int_{0}^{1} \frac{1}{2}(1-x^{2})(1-x^{2}) dx $$ Now, we evaluate the integrals: $$= \frac{1}{2} \int_{0}^{1} (1 - 2x^2 + x^4) dx$$ $$= \frac{1}{2} \left[x - \frac{2}{3}x^3 + \frac{x^5}{5} \right] \Big|_{0}^{1} $$ At \(x = 0\), the result is \(0\) and at \(x = 1\), we get: $$= \frac{1}{2} \left(1 - \frac{2}{3} + \frac{1}{5}\right) $$ $$= \frac{1}{2} \left(\frac{15}{15} - \frac{10}{15} + \frac{3}{15}\right) $$ $$= \frac{1}{2} \cdot \frac{8}{15} $$ Thus, the final answer is: $$\frac{4}{15}.$$

Key concepts

Triple Integrals

Triple integrals extend double integrals into three dimensions. They are used to calculate the volume under a surface defined in three-dimensional space. In mathematical terminology, a triple integral is represented as \( \int \int \int f(x, y, z) \, dx \, dy \, dz \). These integrals can be challenging because they involve multiple layers of integration. To tackle triple integrals, think of stacking layers of double integrals. Each layer corresponds to an axis, with the integral slicing through dimensions one at a time. In many physical applications, triple integrals can represent volumes or accumulations of mass in a defined region. It can be useful to establish a routine approach: treat each integration in turn, simplifying as you go. In our exercise, the triple integral initially involves the nested integration of functions of \( x, y, \) and \( z \). Each layer depends on defining bounds correctly.

Order of Integration

The order of integration in multiple integrals is crucial as it can simplify or complicate the processing of solving an integral. It refers to the sequence in which the integrations are performed. For example, in the original exercise, the target order of integration changed to \( dz \, dy \, dx \) from \( dy \, dz \, dx \). Changing this order requires a thorough understanding of the relationships between the variables and the integration bounds. When choosing an order of integration:

• Identify variable dependencies.
• Assure each variable has well-defined limits that don’t lead to complex functions.
• Simplify final calculations, if possible, by choosing an order that involves simpler algebraic expressions during integration.

Swap orders whenever necessary by considering the existing dependencies and adjusting bounds appropriately. The primary goal is to make calculations straightforward and manageable.

Integration Bounds

Integration bounds specify the range of values for each variable in a multiple integral. They define the limits for each layer of integration. In our exercise, the bounds are given by the equations \( 0 \le x \le 1 \), \( 0 \le y \le \sqrt{1-x^{2}} \), and \( 0 \le z \le \sqrt{1-x^{2}} \). Accurate bounds ensure the region of integration is correctly established. When tackling problems with integration bounds:

• Check dependencies among the variables involved in the function.
• Re-evaluate limits if the order of integration changes, as new variable hierarchies might arise.
• Visualize the limits geometrically when possible—especially if they define three-dimensional objects.

An error in bounding can lead to inaccurate results, making understanding your region’s shape and constraints essential.

Change of Variables

Change of variables is a technique often used to simplify the computation of integrals. This is particularly useful in complex regions where variables have nonlinear interdependencies. By transforming variables, a complex region can become simpler or easier to handle. Common transformations include switching to spherical coordinates or cylindrical coordinates in the context of triple integrals. In practice, change of variables involves:

• Identifying a new variable system that potentially simplifies integration bounds or the integrand.
• Applying Jacobian determinants when transforming volume elements to ensure correct scaling relating to the original variables.
• Reassessing integration bounds based on the new variables.

Successful application of this technique can lead to significantly reduced computational complexity, especially for integrals within geometrically challenging regions.