Question

In Exercises 55 and \(56,\) use a computer algebra system to evaluate the iterated integral. $$ \int_{0}^{1} \int_{y}^{2 y} \sin (x+y) d x d y $$

Short answer

The exact solution to the double integral depends on the computer algebra system used. However, specifying the integral into the computer algebra system and interpreting its result follows the blueprint mentioned above.

Step-by-step solution

Understanding the integral.

This is a double integral of the function \( \sin (x+y) \) . The outer integral is to be taken with respect to \( y \) over the interval [0, 1]. The inner one is with respect to \( x \), bounded by \( y \) and \( 2y \). The order of integrals, as they’re iterated, is important as it can change the result.

Set up the integral in a computer algebra system.

Using a computer algebra system like MATLAB or SageMath, write the double integral as a code. In Sagemath for example, this would look like: f(x,y) = sin(x+y), integral(integral(f, x, y, 2*y), y, 0, 1).

Evaluate the integral.

Now, simply run the code. The computer algebra system does the work of finding the antiderivative and limits by itself, so no further steps are necessary. The output gives the result of the integration.