Ceiling Integration

Calculus Level 3

Calculate $2 \int_{2017}^{1} \big( x - \lceil x \rceil \big) dx.$

Hint: Sketch out the graph.

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The answer is 2016.

1 solution

Andy Hayes
Sep 20, 2016

When the function is graphed, you see right triangles with a base of length $1$ on the $x$ -axis and a height of $1$ below the $x$ axis. There are $2016$ of these triangles in the integral, each with an area of $-\frac{1}{2}$ . Therefore,

$2\int_{2017}^1{\left(x-\lceil x\rceil\right)\ dx}=\boxed{2016}$

Ceiling Integration

The answer is 2016.

1 solution

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