Consider the integral $\int_{0}^{1} \int_{12 x}^{12} f(x, y) d y d x$. Sketch the region of integration and change the order of integration. \[ \begin{array}{l} \int_{a}^{b} \int_{g_{1}(y)}^{g_{2}(y)} f(x, y) d x d y \\ a=\square b=\square \\ g_{1}(y)=\square g_{2}(y)=\square \end{array} \]
Solution
Step 1 ：Consider the integral \(\int_{0}^{1} \int_{12 x}^{12} f(x, y) d y d x\). The limits of y are from 12x to 12, which means y is always greater than or equal to 12x and less than or equal to 12. This gives us the lower and upper bounds for y.
Step 2 ：For x, since y is always greater than or equal to 12x, we can express x as y/12. The lower limit for x is then 0 and the upper limit is y/12.
Step 3 ：The new limits of integration are as follows: The lower limit for y is 0 and the upper limit is 12. The lower limit for x is 0 and the upper limit is y/12.
Step 4 ：The new integral is then \(\int_{0}^{12} \int_{0}^{y/12} f(x, y) dx dy\).
Step 5 ：Final Answer: The new limits of integration are: \(\int_{0}^{12} \int_{0}^{y/12} f(x, y) dx dy\), where a=0, b=12, \(g_{1}(y)=0\), \(g_{2}(y)=y/12\).
