Area Between CurvesJEE Main

Concept

The area trapped between two curves is the integral of their difference over the interval where one lies above the other. Find where they cross, decide which is on top, and integrate top minus bottom.

Key formula

Derivation

Intersections of $f = px^2$ and $g = mx + k$ solve $px^2 - mx - k = 0$, giving limits $x_1, x_2$.

Between them the line is above the parabola, so $A = \int_{x_1}^{x_2}(mx + k - px^2)\,dx = \left[\tfrac{mx^2}{2} + kx - \tfrac{px^3}{3}\right]_{x_1}^{x_2}$.

Scenarios to explore

• Area Between Curves — Definite integral of the gap between a line and a parabola.

Real-world applications

• Work done as the area under a force–displacement graph.
• Probability as area under a density curve.
• Net distance from a velocity–time graph.

JEE exam tips

• Always sketch first to see which curve is on top on each sub-interval.
• For a parabola and a line, the enclosed area has the tidy closed form above.

Common mistakes

• Integrating $f - g$ with the wrong sign, giving a negative 'area'.
• Forgetting to find the intersection points first (wrong limits).
• Missing a sign change where the curves swap top/bottom — split the integral.

Exam traps to avoid

• Area is always positive — take the absolute value or order the functions correctly.
• If the curves don't intersect, there is no finite enclosed region.