Presumably you use Peak Analysis tool to obtain peak area and use Integration gadget to obtain area under curve, basically if the baseline and integration window are set to be same for peak area and curve area, then their values should be the same.
So the curve area is area under the whole curve (above baseline or y=0) and should be equal for all peaks. Peak area is the area under the particular peak. The sum of all peak areas is equal to curve area. Am I right?
"So the curve area is area under the whole curve (above baseline or y=0) and should be equal for all peaks. Peak area is the area under the particular peak. " Correct.
"The sum of all peak areas is equal to curve area. Am I right?" Not necessary. Because x ranges of peaks don't cover the entire curve range. And you can also manually adjust the x range of each peak.