# Change in Consumer Surplus with two goods

Suppose we have two goods, price changes in the two are independent; having seen this question I am considering why the change in total Consumer Surplus is the sum of the change in Consumer Surplus for the two goods separately (if this is true)?

In my reference I see that when we hold $$p_y$$ fixed at $$p_y^0$$ and only $$p_x$$ changes:

$$$$CS = \int_{p^0}^{p^1} x^m (p_x, p_y^0, I^0) dp_x$$$$

Where $$x^m$$ is the Marshallian demand curve for $$x$$.

If we are going to calculate the consumer surplus when both $$p_x$$ and $$p_y$$ change, how should I write the Consumer Surplus equation? Guessing it is one of the following:

$$total \; CS = \int_{p^0}^{p^1} x^m (p_x, p_y^0, I^0) dp_x + \int_{p^0}^{p^1} y^m (p_x^0, p_y, I^0) dp_y$$

or

$$total \; CS = \int_{p^0}^{p^1} \int_{p^0}^{p^1} x^m (p_x, p_y^0, I^0) y^m (p_x^0, p_y, I^0) dp_x dp_y$$