Question

Use a computer algebra system to approximate the iterated integral. $$ \int_{0}^{\pi / 4} \int_{0}^{4} 5 r e^{\sqrt{r \theta}} d r d \theta $$

Short answer

To compute the iterated integral using a computer algebra system, you would first compute the inner integral (with respect to r), treating \(\theta\) as a constant, then compute the outer integral (with respect to \(\theta\)) using the result of the inner integral. The exact result is tool-dependent and usually requires numerical approximation, as the integrals involved are generally not expressible in terms of elementary functions.

Step-by-step solution

Identify the inner integral

The first task is to identify the inner integral, which, in this case, is \(\int_{0}^{4} 5 r e^{\sqrt{r \theta}} d r\). This is the integral one needs to solve first, considering \(\theta\) as a constant.

Compute the inner integral

To compute the inner integral using a computer algebra system (such as Mathematica or Maple), the corresponding command would be something like `integrate(5*r*exp(sqrt(r*theta)), r = 0..4)`. This will calculate the inner integral considering θ as a fixed parameter. The output of this step is some function of \(\theta\).

Compute the outer integral

Once the inner integral has been computed, the next task is to compute the outer integral, which is over \(\theta\). The corresponding command in a computer algebra system would be something like `integrate(inner_integral_output, theta = 0..(pi/4))` where 'inner_integral_output' is replaced with the actual output from step 2. This will compute the outer integral, resulting in the final answer to the problem.

Key concepts

Computer Algebra System

A Computer Algebra System (CAS) is an invaluable tool for mathematicians, engineers, scientists, and students alike. These sophisticated pieces of software can manipulate mathematical expressions in a way that aligns with the symbolic nature of algebra. For example, when faced with an iterated integral such as \[ \int_{0}^{\pi / 4} \int_{0}^{4} 5 r e^{\sqrt{r \theta}} d r d \theta \], a CAS like Mathematica or Maple can be used to perform the heavy lifting.

Using these systems, one can input the integral in a syntax specific to the software. The CAS then applies various algebraic rules and algorithms to compute the result, offering both the exact symbolic answer and a numerical approximation. This functionality is particularly helpful when the integral is too complex to be feasibly calculated by hand, or when an exact analytic solution is challenging to obtain. It is important to note that proficiency with CAS commands and syntax greatly enhances their utility, allowing users to tackle a wider range of problems effectively.

Approximate Integration

Approximate integration is a method used when an integral is either too difficult to solve analytically, or when only a numerical answer is needed. Practical applications often require the numerical evaluation of complicated integrals, like iterated integrals involving exponential functions.

In approximate integration, algorithms such as Simpson's rule, the trapezoidal rule, or Gauss quadrature may be employed to estimate the value of the integral. These methods partition the domain of integration into subintervals, calculate the area under the curve for each subinterval, and sum these areas to approximate the total. When dealing with the iterated integral \[ \int_{0}^{\pi / 4} \int_{0}^{4} 5 r e^{\sqrt{r \theta}} d r d \theta \], numerical techniques can simplify the process by focusing on the numerical evaluation of the inner integral as a function of \theta\, and then performing a numerical integration over \theta\.

Approximate integration provides a valuable means to derive practical answers that, while not exact, can be made increasingly precise by refining the computational methods used.

Calculus

Calculus is the branch of mathematics that deals with rates of change (differential calculus) and the accumulation of quantities (integral calculus). It has applications across science, engineering, economics, statistics, and more, making it one of the most essential areas of mathematics.

When approaching iterated integrals, calculus provides a systematic way to evaluate the accumulated quantity over a region. For example, in the given exercise \[ \int_{0}^{\pi / 4} \int_{0}^{4} 5 r e^{\sqrt{r \theta}} d r d \theta \], differential calculus helps understand how a function changes as its variables change — essential information when setting up an integral. Integral calculus then aids in summing up all the infinitesimal parts to find the whole.

While computing integrals like these can be straightforward for simpler functions, for more complex ones, such as the exponential function dependent on two variables in our iterated integral, the calculations become intricate. This complexity is what often necessitates the use of approximations or a computer algebra system. Understanding the foundational principles of calculus, however, is key to making effective use of these tools and interpreting the results they provide.