Problem 41 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(41-44\) by changing the order of integration in an appropriate way. $$ \int_{0}^{4} \int_{0}^{1} \int_{2 y}^{2} \frac{4 \cos \left(x^{2}\right)}{2 \sqrt{z}} d x d y d z $$

Step-by-Step Solution

Answer

The integral evaluates to \(2 \sin(4)\).

Step 1: Identify the Region of Integration

The given integral is \(\int_{0}^{4} \int_{0}^{1} \int_{2 y}^{2} \frac{4 \cos \left(x^{2}\right)}{2 \sqrt{z}} \, d x \, d y \, d z\). The limits \(x = 2y\) to \(x = 2\) imply that \(x\) varies between the line \(x = 2y\) and the constant \(x = 2\). \(y\) ranges from \(0\) to \(1\) and \(z\) from \(0\) to \(4\).

Step 2: Graphical Representation of Region

Visualize the region of integration in the \(xy\)-plane: it is bounded by \(x = 2y\), \(x = 2\), and \(y = 0\) to \(y = 1\). Considering \(z\) from \(0\) to \(4\) stacks this region uniformly in the \(z\)-axis.

Step 3: Change the Order of Integration

To integrate with respect to \(z\) first, observe that for fixed \(y\) and \(z\), \(x\) spans from \(2y\) to \(2\). Reverse the order: \(z\) from \(0\) to \(4\), then*\(y\) from \(0\) to \(\frac{x}{2}\), and \(x\) from \(0\) to \(2\).

Step 4: Set up the New Integral Order

Rewrite the integral as \(\int_{0}^{2} \int_{0}^{x/2} \int_{0}^{4} \frac{4 \cos(x^2)}{2 \sqrt{z}} \, dz \, dy \, dx\).

Step 5: Perform the Integration with Respect to \(z\)

Integrate the innermost integral: \(\int_{0}^{4} \frac{4 \cos(x^2)}{2 \sqrt{z}} \, dz\). This becomes \(4 \cos(x^2) \left[ \sqrt{z} \right]_{0}^{4} = 4 \cos(x^2) (2 - 0) = 8 \cos(x^2)\).

Step 6: Integrate with Respect to \(y\)

Integrate \(\int_{0}^{x/2} 8 \cos(x^2) \, dy\). This becomes \(8 \cos(x^2) \left[ y \right]_{0}^{x/2} = 8 \cos(x^2) \left( \frac{x}{2} \right) = 4x \cos(x^2)\).

Step 7: Final Integration with Respect to \(x\)

Integrate \(\int_{0}^{2} 4x \cos(x^2) \, dx\). Let \(u = x^2\), then \(du = 2x \, dx\) or \(x \, dx = \frac{1}{2} \, du\). Change limits: when \(x = 0\), \(u = 0\); when \(x = 2\), \(u = 4\).

Step 8: Substitute and Solve the Integral

Substitute to get \(\int_{0}^{4} 2 \cos(u) \, du = 2 [\sin(u)]_{0}^{4}\). Evaluate \(2 (\sin(4) - \sin(0)) = 2 \sin(4)\).

Key Concepts

Triple Integrals

Triple integrals are a powerful tool employed in calculus to calculate the volume under a surface or compute quantities within a three-dimensional region. Essentially, they extend the concept of finding the area under a curve (single integral) and the volume under a surface (double integral) to three dimensions.

When evaluating a triple integral, we have three nested layers of integration, typically written in the form: \[\int \int \int f(x, y, z) \, dx \, dy \, dz\]This involves integrating a function over a solid region in three-dimensional space. The limits of each integral can be fixed (constant) or variable, depending on the particular problem.

Each of the three integrals corresponds to an axis in a 3D coordinate system (often Cartesian coordinates). Therefore, the order of integration, which variable you integrate first, can be varied and sometimes significantly simplify the computational load. This is crucial for solving complex integrals as seen in our problem, by initially integrating with respect to \(z\) to reduce complexity.

Integration Techniques

Integration techniques are methods used to solve integrals efficiently and accurately. These techniques become essential, especially when dealing with complex or non-standard functions in calculus.

Common integration methods used include:

• Substitution: This is similar to applying the chain rule in reverse to simplify the integral using an intermediate variable.
• Integration by Parts: Used when products of functions are involved, applying the rule \(\int u \, dv = uv - \int v \, du\).
• Partial Fraction Decomposition: Used primarily for rational functions, breaking them into simpler fractions that are easier to integrate.

In the provided exercise, substitutions were specifically useful. By simplifying the expression from \(x \, dx = \frac{1}{2} \, du\), it allowed for easier evaluation of the integral by transforming areas that were intricate into simpler expressions. This approach reduces errors and gives coherent results more efficiently.

Changing Variables

Changing variables is an integration strategy that involves altering the order or type of variables to simplify the evaluation of an integral. This approach is a feature in multivariable calculus and is pivotal for solving integrals like those presented in the exercise.

In practice, changing variables often leads to reversing the integration order, which can make the computation more manageable. For example, in our problem, instead of integrating first over \(x\), we switched to integrate over \(z\).

• This transformation simplified the region of integration and reduced complexity.
• Re-organizing the limits according to the new integration order helped us to gain a clearer perspective on which limits affect which variables directly.

Moreover, converting variables through substitution, like setting \(u = x^2\), facilitated changing how we approach specific integrals or shifted limits to a more comfortable place to compute.

This technique highlights the importance of adaptable thinking in calculus, where understanding the relationships between variables can reduce a seemingly complex integral into a straightforward calculation.