Example 1

**Input list of rectangles**: [0, 1, 2, 3, 4, 5]

**Input rectangle width**: 1.5

**Output area**: 22.5

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Definite integrals

**You will learn about**:
numerical integration.

The simple problem here is to sum the area of a bunch of rectangles with different heights but the same width. That's all you need to know. But if you don't mind learning a bit of calculus, read on.

The slightly longer version is that we're trying to calculate the area under a complicated line or curve (the black line in the figure below). One way to do this is to draw rectangles under the curve and just add up the areas of all these rectangles. You can choose the rectangles in a few different ways but the idea is that if you use more and more rectangles, you'll get a more accurate answer (see figure caption).

If the black curve is described by a function $f(x)$ then the area under the curve is the definite integral of $f$ between some endpoints $a$ and $b$ $$ \int_a^b f(x) \, dx \approx \sum_{k=0}^N f(x_i) \Delta x $$ where we decided to use $N$ rectangles each of width $\Delta x$. The height of the $i^\textrm{th}$ rectangle is $f(x_i)$ which can be chosen in a few different ways but is given in this problem.

**Input**:
A list of rectangle heights (negative heights give a rectangle "negative area"), and a rectangle width.

**Output**:
The sum of the rectangle areas.

Difficulty | Timesink | ||
---|---|---|---|

Maximum runtime | 60 s | Max. memory usage | 250 MiB |

Function signature | area_of_rectangles(rectangle_heights, rectangle_width) |

Write a function that accepts the input as function parameters and returns the correct output. Make sure to read the description above to produce the correct output in the correct format and use the correct function signature so we can run your code. A good first step is to try reproducing the example(s). Your code must not take longer than the maximum runtime to run and must not use more memory than the allowed limit.

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- Thanks to rmueller for suggesting this problem!
- The method we used to compute the area under the curve here using rectangles is called a Riemann summation. You could use more complicated shapes: The trapezoidal rule uses trapezoids and Simpson's rule uses quadratic polynomials.
- Instead of using upright rectangles like we did here with the Riemann sum, you can use rectangles placed sideways which is kind of how the area under the curve is interpreted with Lebesgue integration.

Let us know what you think about this problem! Was it too hard? Difficult to understand? Also feel free to
discuss the problem, ask questions, and post cool stuff on Discourse. You should be able see a discussion
thread below. Feel free to post your solutions but if you do please **organize and document your code
well** so others can learn from it.