10.3. Step-by-Step Procedure

Try these first

1. 1.

   Integrate $\frac{\partial z}{\partial x} = 5x$.

   Hint

   Remember to treat y as a constant.

2. 2.

   What is the result of integrating $\frac{\partial z}{\partial y} = 3y^2$?

   Hint

   Think about what happens to x during this integration.

3. 3.

   What is direct integration used for?

   • Solving ordinary differential equations
   • Solving Partial Differential Equations
   • Finding limits of functions

   Hint

   Think about which type of equations involve partial derivatives.

4. 4.

   When given $\frac{\partial z}{\partial x} = f(x,y)$, we integrate with respect to which variable?

   • True
   • False

   Hint

   This ensures we correctly find \\( z \\) as a function of both variables.

5. 5.

   Given the PDEs $\frac{\partial z}{\partial x} = 3x^2$ and $\frac{\partial z}{\partial y} = 4y$, derive the function z.

   Hint

   Remember to add both arbitrary functions after integration.

6. 6.

   For the PDE $\frac{\partial z}{\partial x} = sin(xy)$, integrate to find z where y is treated as constant.

   Hint

   Recall how we adjust our integration based on treating y as a constant.