Step 1 - Sketch the Region: Draw rough graphs of all given curves. Identify the enclosed region visually. This prevents errors in choosing limits and determining which curve is on top.

Step 2 - Find Intersection Points: Solve the equations of the curves simultaneously to find all intersection points. These are your limits of integration. For transcendental equations, use numerical or graphical methods.

Step 3 - Determine Upper/Lower $\frac{or Right}{Left}$ Curves: In each subinterval between consecutive intersection points, determine which curve has the larger y-value (for dx integration) or larger x-value (for dy integration) by substituting a test point.

Step 4 - Choose the Variable of Integration: If both curves are easily expressed as y = f(x), use vertical strips (dx). If they're naturally x = g(y), use horizontal strips (dy). Choose whichever gives fewer integral pieces.

Step 5 - Exploit Symmetry: Check if the region has symmetry about the x-axis, y-axis, origin, or y = x. If so, compute the area of the symmetric part and multiply by the appropriate factor.

Step 6 - Set Up and Evaluate the Integral: Write the integral as (upper - lower) dx or (right - left) dy. Split at all crossing points. Evaluate each piece using standard integration techniques.

Step 7 - Verify the Answer: Check that the area is positive. Compare with rough geometric estimates (e.g., the area should be less than the bounding rectangle). Verify dimensions and units.

Common verification: the area between a parabola and a chord is always $\frac{2}{3}$ of the circumscribing parallelogram. The area between y = $x^2$ and y = x should be less than the 1x1 square enclosing it (1/6 < 1, checks out).