Answer
$\int_{1}^{1} \sqrt{1+x^4}~dx = 0$
Work Step by Step
Note that the interval $1 \leq x \leq 1$ has a width of zero. If we interpret the integral as the area under the graph, then the area must be zero since the width of the interval is zero.
Therefore:
$\int_{1}^{1} \sqrt{1+x^4}~dx = 0$