Why is the marginal cost curve not the same as the supply curve?

My personal explanation is that since the marginal cost curve is cost of producing each new product, the supply curve just represents the break even point of selling a certain number of these products. The seller can sell some at a profit, and some at a loss, and as long as the loss and profit cancel out, he's on the supply curve. Thus the supply of producing x goods is just the average of the marginal cost curve from 0 to x.

This explanation seems logical to me, but I have been given an alternative explanation - the supply curve is that of the market, whereas the marginal cost curve is intrinsic to the firm. However, I am unsure as to why the market supply would be related to the marginal cost curve as the average of its integral.