Answer:
The part of the sentence that is the dangling modifier is:
"Swinging through the air".
Step-by-step explanation:
A dangling modifier is a word or phrase that is placed incorrectly in a sentence, which creates ambiguity. Its position makes it unclear to what or whom it refers.
In the sentence "Swinging through the air, the children were delighted by the acrobats on the stage," it is unclear who was swinging through the air. We can easily assume it was the acrobats, because that is what acrobats do. However, when we first read the sentence, we may also easily think it refers to the children, because "children" is the noun closer to the modifier. Thus, we can say that "swinging through the air" is the dangling modifier.