IntegralGrader is a specialized grading class used to grade the construction of integrals. Students can input the limits on the integral, the variable of integration, and the integrand (or some subset thereof), and will be graded correct if their construction is numerically equivalent to the instructor's construction. This method of grading allows for arbitrary variable substitutions and redefinitions.

The grader numerically evaluates the student- and instructor-specified integrals using scipy.integrate.quad. This quadrature-based integration technique is efficient and flexible. It handles many integrals with poles in the integrand and can integrate over infinite domains.

However, some integrals may behave badly. These include, but are not limited to, the following:

• integrals with highly oscillatory integrands
• integrals that evaluate analytically to zero

In some cases, problems might be avoided by using the integrator_options configuration key to provide extra instructions to scipy.integrate.quad, as documented below.

## XML Setup #

We recommend copying the following XML to set up a problem using IntegralGrader:

<style>
.xmodule_display.xmodule_CapaModule .problem .capa_inputtype.textline input {
min-width: 0 !important;
}
.xmodule_display.xmodule_CapaModule div.problem section div span.MathJax {
display: inline-block !important;
}
.xmodule_display.xmodule_CapaModule div.problem section div span.MathJax_Preview {
display: inline-block !important;
}
</style>

<span>
<table>
<col style="width:10%"/>
<col style="width:90%"/>
<tbody>
<tr>
<td colspan="2">
</td>
</tr>
<tr>
<td>
<p> $$\displaystyle \huge{ \int }$$</p>
</td>
<td>
<br/>
<textline inline="1" size="10" correct_answer="e^x" trailing_text="  [mathjaxinline] dx [/mathjaxinline]"/>
</td>
</tr>
<tr>
<td colspan="2">
</td>
</tr>
</tbody>
</table>
</customresponse>
</span>

This sets up an integral where students can input the limits of integration and the integrand (the variable of integration has been fixed to be x in this case).

Further examples of formatting integrals are shown in the example course.

## Specifying the Input Format #

The grader must be told which input is what, based on the order that the inputs appear in the XML. This is done through the input_positions dictionary. If not specified, it is assumed that the following positions are used:

input_positions = {
'lower': 1,
'upper': 2,
'integrand': 3,
'integration_variable': 4
}

This requires students to enter all four parameters in the indicated order.

If the author overrides the default input_positions value, any subset of the keys ('lower', 'upper', 'integrand', 'integration_variable') may be specified. Key values should be

• continuous integers starting at 1, or
• (default) None, indicating that the parameter is not entered by student

For example,

input_positions = {
'lower': 1,
'upper': 2,
'integrand': 3
}

indicates that the problem has 3 input boxes which represent the lower limit, upper limit, and integrand in that order. The integration_variable is NOT entered by student and is instead given by the value specified by author in 'answers'.

Here is a sample grader for the above XML:

>>> from mitxgraders import *
...         'lower':'0',
...         'upper':'1',
...         'integrand':'e^x',
...         'integration_variable':'x'
...     },
...     input_positions = {
...         'upper': 1,
...         'integrand': 2,
...         'lower': 3
...     }
... )

Note that when students specify their own variable of integration, it must not conflict with a variable already present in the problem.

The author's answer should be specified as a dictionary with the following keys:

answers = {
'lower': 'lower_limit',
'upper': 'upper_limit',
'integrand': 'integrand',
'integration_variable': 'variable_of_integration'
}

Note that each entry is a string value.

IntegralGrader can handle integrals over both finite and infinite domains. A special constant 'infty' is recognized to cater for the infinite case (and takes on the special value float('inf')).

## Other Options #

If you wish to allow a student's integrand to be complex-valued at any point in the domain of the integral, set complex_integrand=True. If set to False (the default), a student's submission will be graded as incorrect if their integrand becomes complex anywhere in the domain.

You can modify the integration options used by scipy.integrate.quad by passing a dictionary of keyword-argument values using the option integrator_options.

The following options from FormulaGrader are available for use in IntegralGrader:

• user_constants
• user_functions
• whitelist
• blacklist
• tolerance
• samples (default: 1)
• variables
• sample_from
• failable_evals

Unless otherwise specified, the defaults are the same as in FormulaGrader.

## Option Listing #

Here is the full list of options specific to an IntegralGrader.

grader = IntegralGrader(
input_positions=dict,
)