Problem 15 - Calculus Early Transcendentals

Question

Evaluate the following integrals in cylindrical coordinates. The figures illustrate the region of integration. $$\int_{0}^{2 \pi} \int_{0}^{1} \int_{-1}^{1} d z r d r d \theta$$

Step-by-Step Solution

Answer

Answer: The final answer to the integral is 2π.

Step 1: Integrate with respect to z

Integrate the given function with respect to z: $$\int_{-1}^1 dz = z$$ Evaluate this function from z = -1 to z = 1: $$[(1)-(-1)] = 2$$ Now our integral becomes: $$2 \int_{0}^{2 \pi} \int_{0}^{1} r dr d\theta$$

Step 2: Integrate with respect to r

Integrate the function with respect to r: $$\int_{0}^{1} r dr = \frac{1}{2}r^2$$ Evaluate this function from r = 0 to r = 1: $$\left[\frac{1}{2}(1)^2\right] - \left[\frac{1}{2}(0)^2\right] = \frac{1}{2}$$ Now our integral becomes: $$2 \left(\frac{1}{2}\right) \int_{0}^{2 \pi} d\theta$$

Step 3: Integrate with respect to θ

Integrate the function with respect to θ: $$\int_{0}^{2 \pi} d\theta = \theta$$ Evaluate this function from θ = 0 to θ = 2π: $$[(2\pi)-(0)] = 2\pi$$

Step 4: Evaluate the final result

Multiply all the constants together to obtain the final result: $$2 \left(\frac{1}{2}\right)(2\pi) = (1)(2\pi) = 2\pi$$ The final result for the integral is: $$\int_{0}^{2 \pi} \int_{0}^{1} \int_{-1}^{1} d z r d r d \theta = 2\pi$$

Key Concepts

Triple Integral

Triple integrals are a powerful tool used in multivariable calculus to calculate the volume under a surface in three dimensions. Essentially, a triple integral expands on the concept of a double integral by adding an additional layer of integration. When dealing with triple integrals, you integrate a function with respect to three different variables. In the case of cylindrical coordinates, these variables are usually written as \(z\), \(r\) (the radial distance from the origin), and \(\theta\) (the angle around the origin). For each variable, integration is performed over a specified range.
Given the function you are integrating, you perform successive integrations with respect to each variable, typically starting with the innermost integral. In our provided example, the order is \(dz\), \(dr\), then \(d\theta\). This order can vary depending on the problem. Each layer of the integral essentially reduces the dimensions of the region being considered:

• The innermost integral (with respect to \(z\)) removes one dimension.
• The next one (with respect to \(r\)) compresses it further.
• The outermost (with respect to \(\theta\)) evaluates the shape or region completely.

This systematic process results in a scalar value, representing the total accumulation across the three dimensions.

Integration

Integration is a fundamental operation in calculus and refers to the process of finding the integral or antiderivative of a function. It is commonly used to calculate areas, volumes, and other quantities that accumulate over a range. In the context of the provided exercise, integration is performed in three steps, each focusing on a different variable as per cylindrical coordinates: \(z\), \(r\), and \(\theta\).

Each integration step simplifies the expression:

• Integrating with respect to \(z\): This calculates the change along the vertical axis within the specified limits (-1 to 1). The given formula produces an intermediate result of 2, framing the problem for the next step.
• Integrating with respect to \(r\): This step considers the radial part, essentially calculating how the function behaves as you move outward from the origin. Evaluating this from 0 to 1 provides \(\frac{1}{2}\).
• Integrating with respect to \(\theta\): This determines the integral around the circle in the \(xy\)-plane, resulting in \(2\pi\), or a full rotation.

Ultimately, integration is about accumulation, and each integration step sums up small pieces to form the complete volume under consideration in the exercise.

Coordinate Systems

Understanding different coordinate systems is essential in calculus as it allows visualization and tackling problems from different angles. In the original exercise, the use of cylindrical coordinates is crucial to simplify the integration. Cylindrical coordinates are a system that extend polar coordinates into three dimensions, and they are particularly useful when dealing with problems that have symmetry about an axis. They are defined by:

• \(r\): Radial distance from the origin projected onto the \(xy\)-plane.
• \(\theta\): The angle between the positive \(x\)-axis and the line segment from the origin to the point's projection.
• \(z\): Elevation or height from the \(xy\)-plane.

This system is excellent for modeling cylindrical and rotationally symmetric objects, making the integration process straightforward. It allows for the simplification of the integration bounds and in essence, transforms a complex geometric problem into a simpler one.
By changing the traditional Cartesian axes into cylindrical coordinates, the integration over a 3D volume becomes much more manageable, particularly when the region of integration is inherently circular or cylindrical.