Problem 39 - Calculus Early Transcendentals

Question

Rewrite the following integrals using the indicated order of integration and then evaluate the resulting integral. $$\int_{0}^{5} \int_{-1}^{0} \int_{0}^{4 x+4} d y d x d z \text { in the order } d z d x d y$$

Step-by-Step Solution

Answer

Answer: The result of the triple integral is \(\frac{2}{3}\).

Step 1: Understand the given integral

The given integral is $$\int_{0}^{5} \int_{-1}^{0} \int_{0}^{4 x+4} d y d x d z$$ with the order of integration being \(dy \, dx \, dz\).

Step 2: Find the new limits and write the new integral

To find the limits for the new order of integration (\(dz \, dx \, dy\)), we must write x, y, and z bounds in terms of each other: From the bounds of \(x\): $$ -1 \leq x \leq 0 $$ From the bounds of \(y\): $$ 0 \leq y \leq 4x + 4 $$ From the z bounds: $$ 0 \leq z \leq 5 $$ To express \(x\) and \(y\) in terms of \(z\), we need to solve for \(x\) from the \(y\) bound: $$ y = 4x + 4 ~\Rightarrow~ x = \frac{(y - 4)}{4} $$ Now we can write the integral with the new order: $$ \int_{0}^{5} \int_{-1}^{\tfrac{(y - 4)}{4}} \int_{0}^{y} dz \, dx \, dy $$

Step 3: Solve the new integral

Now we can solve the triple integral step by step: 1. Integrate w.r.t \(z\): $$ \int_{0}^{5} \int_{-1}^{\tfrac{(y - 4)}{4}} y \, dx \, dy $$ since \(\int_{0}^{y} dz = y\). 2. Integrate w.r.t \(x\): $$ \int_{0}^{5} \left[y \left(\frac{(y - 4)}{4} + 1\right)\right] dy $$ since \(\int_{-1}^{\frac{(y - 4)}{4}} dx = \frac{(y - 4)}{4} + 1\). 3. Integrate w.r.t \(y\): \begin{align*} &\int_{0}^{5} y \left(\frac{(y - 4)}{4} + 1\right) dy \\ &= \int_{0}^{5} \left(\frac{1}{4}y^2 - \frac{3}{4}y \right) dy \\ &= \left[\frac{1}{12}y^3 - \frac{3}{8}y^2\right]_0^5 \\ &= \left(\frac{1}{12}(5)^3 - \frac{3}{8}(5)^2\right) - 0 \\ &= \frac{125}{12} - \frac{75}{8}\\ &= \frac{2}{3} \end{align*} The final result of the triple integral is \(\frac{2}{3}\).

Key Concepts

Order of Integration

The order of integration in a triple integral refers to the sequence in which the integrations are carried out. In multiple integrals, changing the order of integration can simplify the problem or make it possible to solve. When tackling such problems, always check the

• Order of the original integral.
• Order you need to rearrange it into.

For the given exercise, the original order was \[ \int_{0}^{5} \int_{-1}^{0} \int_{0}^{4x+4} dy \, dx \, dz \]which is \( dy \, dx \, dz\).
We needed to rewrite the integral with the order \( dz \, dx \, dy \),showing each integration step intricately influences the subsequent step. Changing the order may mean redefining variable limits based on the new sequence. Thus, it’s imperative to first understand the relationship among variables and adjust the limits accordingly.

Limits of Integration

Limits of integration specify the range over which each variable varies in the integration process. Re-calculating these limits is crucial when you change the order of integration. This involves expressing one variable in terms of others or identifying a new range that maintains the relationship of the variables.
In our exercise, the original bounds were:

• \( x: -1 \leq x \leq 0 \)
• \( y: 0 \leq y \leq 4x + 4 \)
• \( z: 0 \leq z \leq 5 \)

To find the new limits for the order \( dz \, dx \, dy \):

• We solve \( y = 4x + 4 \) for \( x \), yielding \( x = \frac{(y - 4)}{4} \).
• Using this, bounds for \( x \) change to \(-1 \leq x \leq \frac{(y - 4)}{4} \).
• So the new integral becomes \[ \int_{0}^{5} \int_{-1}^{\tfrac{(y-4)}{4}} \int_{0}^{y} dz \, dx \, dy \].

Redefining limits correctly ensures that the geometric region of integration remains constant, thereby preserving the integrity of the calculation.

Iterated Integrals

Iterated integrals involve integrating one variable at a time in a specified order. Each iteration simplifies the integral step-by-step, moving closer to the result. Understanding iterated integrals requires identifying the integration target first and ensuring each layer of integration follows the appropriate bounds.
In the provided solution:

• The first step was integrating with respect to \( z \): \[ \int_{0}^{y} dz = y \].
• The solution then integrated with respect to \( x \), yielding \[ y \left(\frac{(y - 4)}{4} + 1\right) \].
• Finally, integrating with respect to \( y \), it resulted in \[ \int_{0}^{5} \left(\frac{1}{4}y^2 - \frac{3}{4}y\right) dy \] where the operation completed as \( \frac{2}{3} \).

The importance of iterated integrals is they transform a complex integration process into systematic steps, helping to gradually resolve intricate mathematical relationships in triple integrals.