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Would someone clarify this for me... Is antiderivatives just another name for intergral and why is intergral of a function is the area under the curve?

Antiderivatives is another word for integral.

Why area? THink on this...the integral is the sum of all the "heights" times width (dx). Take a look at this depiction of the trapezoidal rule:

The anti-derivative is another name for the indefinite integral. The term integral is ambiguous here, we have both definite and indefinite integrals.
The definite integral is the area bounded by the function, the x-axis and the limits of integration. The last part of your question is slightly confusing. The definite integral is defined as the limit of a sum which represents the area under the function.

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