Question
Find the area enclosed between the parabola and the line .
Solution — Step by Step
Set :
So the curves meet at and .
Pick a test point in between, say . Line: . Parabola: . So the line is above the parabola throughout the interval .
At : .
At : .
Area = square units.
Why This Works
To find area between two curves, integrate (upper − lower) with respect to . The bounds are the -coordinates of intersection. If the upper-lower relation flips, split the integral at the crossover point and integrate each piece separately.
The intuition: at each in the interval, the vertical strip has height (upper − lower) and width , so its area is the integrand. Adding all strips via integration gives the total enclosed area.
Alternative Method
Integrate with respect to instead — invert both curves: for the parabola, for the line. Set up two horizontal-strip integrals. More work in this case, useful when the curves are easier to express in form.
Common Mistake
Students forget to verify which curve is on top. If you integrate (parabola − line) instead of (line − parabola), you get — area can never be negative. Either check at a test point first, or take the absolute value at the end. Also: never just compute blindly without knowing the order — the absolute value handles sign but you still need the right order to set it up.