Integration with Initial Conditions

[Diagram: math-integration-area] - Available in full study mode

Area between two curves — method: \text(Area) = \int_a^b [f(x) - g(x)] dx where a and b are the x-coordinates of the intersection point. Step 1: Find intersections (solve f(x) = g(x)) Step 2: Identify which curve is on top Step 3: Integrate [top − bottom] between the limits

Why we subtract the curves: Think of area between curves as many thin vertical strip. Each strip has height [top − bottom] and width dx. Adding all strip: Area = ∫[top − bottom] dx. The subtraction ensures we get the gap between the curve, not the full area under each.

When curves cro — split the integral: If the curves switch which is on top partway through the interval, you must split the integral at the cro ing point. Area = \int_a^c [f(x) - g(x)] dx + \int_c^b [g(x) - f(x)] dx where c is the point where f(x) = g(x) inside the interval.

Don't add negative values: If you integrate over an interval where the 'wrong' function is on top, the result is negative. Area is always positive. Take the absolute value of each sub-integral and then add them. On GDC: use ∫|f(x) − g(x)| dx to get total area directly.

Show-your-setup marks: On IB mark scheme, one mark is awarded specifically for: • Writing the correct integral expre ion \int_a^b [f - g] dx Even if you make arithmetic errors later, you can still earn this mark. Always write out the integral explicitly before evaluating.

Area problems with GDC (Paper 2 strategy): On Paper 2: 1. Plot both functions on GDC 2. Use 'intersection' tool to find limit 3. Use definite integral function: ∫(top − bottom) between limit You must still show the setup (the integral expre ion) to earn method mark.