JEE · Mathematics

Area Under Curves

Use definite integration to find nonnegative geometric areas bounded by simple curves by locating intersections, deciding which curve is above or to the right, and splitting intervals whenever sign or curve order changes.

Subject: Mathematics
Syllabus unit: Integral Calculus

Explicit in both JEE Main 2026 Unit 8 and JEE Advanced 2026 Integral Calculus
Geometric area is always nonnegative, unlike a signed definite integral
No invented weightage, question counts or trend percentages

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In short

A definite integral is signed, but geometric area is nonnegative.

Before integrating, find relevant intersections and determine which curve is above the other on each interval. If a curve crosses the axis, or two curves exchange order, split the interval. Geometric area is obtained from the magnitude of the vertical or horizontal difference, not by accepting a negative signed integral as an area.

Syllabus mapping

Syllabus mapping
Unit	Topics
Integral Calculus	Areas bounded by simple curves, Application of definite integrals to area determination

Unit
Integral Calculus
Topics
Areas bounded by simple curves, Application of definite integrals to area determination

What this chapter contains and why it matters

What this chapter contains and why it matters
Question	Direct answer
What is the chapter about?	Using definite integrals to compute nonnegative geometric area bounded by simple curves, with intersections and interval splitting.
What is the central method choice?	Locate intersections, determine curve order on each interval, split where order changes, and integrate the nonnegative difference.
Where do most mistakes begin?	Reporting a negative signed integral as area, missing an intersection, and using top-minus-bottom after curves exchange order.
What should come before Area Under Curves?	Definite Integrals for fixed-limit evaluation, and Coordinate Geometry for intersection and order reasoning.
What comes after it?	Differential Equations continues the integral-calculus sequence with equations relating functions and derivatives.

Question
What is the chapter about?
Direct answer
Using definite integrals to compute nonnegative geometric area bounded by simple curves, with intersections and interval splitting.
Question
What is the central method choice?
Direct answer
Locate intersections, determine curve order on each interval, split where order changes, and integrate the nonnegative difference.
Question
Where do most mistakes begin?
Direct answer
Reporting a negative signed integral as area, missing an intersection, and using top-minus-bottom after curves exchange order.
Question
What should come before Area Under Curves?
Direct answer
Definite Integrals for fixed-limit evaluation, and Coordinate Geometry for intersection and order reasoning.
Question
What comes after it?
Direct answer
Differential Equations continues the integral-calculus sequence with equations relating functions and derivatives.

Official JEE syllabus documents define content scope. They do not publish chapter weightage, so none is asserted here.

Official JEE syllabus mapping for Area Under Curves

Verified against the current JEE Main 2026 syllabus and JEE Advanced 2026 syllabus.

Official JEE syllabus mapping for Area Under Curves
Concept group	JEE Main 2026	JEE Advanced 2026	Preparation note
Area determination	Explicitly includes determining areas of regions bounded by simple curves in standard forms.	Explicitly includes application of definite integrals to determination of areas bounded by simple curves.	Both papers restrict scope to simple bounded curves; do not expand into unsupported complex-region theory.

Concept group
Area determination
JEE Main 2026
Explicitly includes determining areas of regions bounded by simple curves in standard forms.
JEE Advanced 2026
Explicitly includes application of definite integrals to determination of areas bounded by simple curves.
Preparation note
Both papers restrict scope to simple bounded curves; do not expand into unsupported complex-region theory.

Sources: official Main Unit 8 area scope and official Advanced Integral Calculus area scope, linked in the sources section below.

Before this chapter

Prerequisites: what you should know before Area Under Curves

Prerequisites: what you should know before Area Under Curves
Prerequisite	You are ready if you can…	If not, repair this first
Definite integrals	Evaluate ∫_a^b f(x)dx with the Fundamental Theorem.	Revise Definite Integrals.
Graphing simple functions	Sketch a simple polynomial or trigonometric curve.	Revise function graphs.
Coordinate intersections	Solve two equations simultaneously for intersection points.	Revise Coordinate Geometry.
Function sign analysis	Determine where a function is positive or negative on an interval.	Revise sign-chart methods.

Prerequisite
Definite integrals
You are ready if you can…
Evaluate ∫_a^b f(x)dx with the Fundamental Theorem.
If not, repair this first
Revise Definite Integrals.
Prerequisite
Graphing simple functions
You are ready if you can…
Sketch a simple polynomial or trigonometric curve.
If not, repair this first
Revise function graphs.
Prerequisite
Coordinate intersections
You are ready if you can…
Solve two equations simultaneously for intersection points.
If not, repair this first
Revise Coordinate Geometry.
Prerequisite
Function sign analysis
You are ready if you can…
Determine where a function is positive or negative on an interval.
If not, repair this first
Revise sign-chart methods.

This is a readiness check, not a weightage or scoring-priority list.

Concepts in this chapter

1. Identify all region boundaries

A correct integral setup starts from a clear description of the bounded region.

Identify all curves and lines forming the boundary of the region before setting up any integral.

2. Solve intersections before setting limits

Limits of integration come from intersection points, not assumption.

Solve for intersection points of the bounding curves before choosing the limits of integration.

3. Test which curve is above or rightmost

Order must be tested, not assumed from the equations alone.

Test which curve is above on each x-interval, or which curve is rightmost on each y-interval, using a sample point.

4. Split wherever ordering changes

A single integral across an order change gives an incorrect area.

Split the interval wherever the curve ordering changes, and integrate each piece separately.

5. Integrate top minus bottom or right minus left

Always integrate the nonnegative difference on each piece.

Integrate top minus bottom, or right minus left, ensuring the integrand is nonnegative on each piece.

6. Add nonnegative piece areas and sanity-check

Sanity-check the total against the sketch and a rough bounding rectangle.

Add the nonnegative areas of all pieces, then sanity-check the total against the sketch and a rough bounding rectangle.

Method selector for area problems

Match the region signal to the correct first approach.

Method selector for area problems
Region signal	First approach
One curve and x-axis, no sign change	Direct nonnegative integral
Curve crosses x-axis	Locate zeros and split
Two curves	Solve intersections, determine top/bottom
Vertical representation becomes multi-branch	Consider horizontal slicing
Symmetric region	Use symmetry only after geometric and integrand validation
Closed region unclear from algebra	Sketch before integrating

Region signal
One curve and x-axis, no sign change
First approach
Direct nonnegative integral
Region signal
Curve crosses x-axis
First approach
Locate zeros and split
Region signal
Two curves
First approach
Solve intersections, determine top/bottom
Region signal
Vertical representation becomes multi-branch
First approach
Consider horizontal slicing
Region signal
Symmetric region
First approach
Use symmetry only after geometric and integrand validation
Region signal
Closed region unclear from algebra
First approach
Sketch before integrating

Worked reasoning: area between y=x and y=x²

Find the area between y=x and y=x² on [0,1]

Known: intersections satisfy x=x², so x=0 or x=1.

Method choice: on (0,1), x>x², so the line is above the parabola.

A = ∫_0^1 (x-x²)dx = [x²/2-x³/3]_0^1 = 1/2-1/3 = 1/6.

Validity check: the integrand x-x²=x(1-x) is nonnegative on [0,1], so the computed area is positive and no further split is required.

Area formulas and their conditions

Area formulas and their conditions
Record	Expression	Conditions	Trap	Validation
Area above x-axis	A = ∫_a^b f(x)dx	f(x) ≥ 0 on [a,b] and integrable	Using when f changes sign	Sign chart
General axis area	A = ∫_a^b \|f(x)\|dx	f integrable; split at zeros for practical evaluation	Dropping absolute value	Area nonnegative
Area between two curves	A = ∫_a^b [f(x)-g(x)]dx	f(x) ≥ g(x) throughout interval	Assuming order without check	Test point/order
General between-curves area	A = ∫_a^b \|f(x)-g(x)\|dx	Integrable; practical splitting at intersections/order changes	One signed integral across crossings	Intersection list
Horizontal slicing	A = ∫_c^d [x_right(y)-x_left(y)]dy	Region is correctly represented horizontally	Forcing dx when functions are multivalued in x	Sketch

Record
Area above x-axis
Expression
A = ∫_a^b f(x)dx
Conditions
f(x) ≥ 0 on [a,b] and integrable
Trap
Using when f changes sign
Validation
Sign chart
Record
General axis area
Expression
A = ∫_a^b |f(x)|dx
Conditions
f integrable; split at zeros for practical evaluation
Trap
Dropping absolute value
Validation
Area nonnegative
Record
Area between two curves
Expression
A = ∫_a^b [f(x)-g(x)]dx
Conditions
f(x) ≥ g(x) throughout interval
Trap
Assuming order without check
Validation
Test point/order
Record
General between-curves area
Expression
A = ∫_a^b |f(x)-g(x)|dx
Conditions
Integrable; practical splitting at intersections/order changes
Trap
One signed integral across crossings
Validation
Intersection list
Record
Horizontal slicing
Expression
A = ∫_c^d [x_right(y)-x_left(y)]dy
Conditions
Region is correctly represented horizontally
Trap
Forcing dx when functions are multivalued in x
Validation
Sketch

Common mistakes and what they actually indicate

Reporting a negative definite integral as geometric area.
Knowledge gap
Why it happens
Geometric area is nonnegative by definition; a negative signed integral indicates the curve lies below the axis on that interval.
How it is corrected
Take the magnitude of the signed integral, or split and integrate |f(x)-g(x)|, to get geometric area.
Failing to find all intersections.
Execution error
Why it happens
Missing an intersection point produces incorrect limits or a missed order change.
How it is corrected
Solve the intersection equation completely before setting up limits.
Using top-minus-bottom after the curves exchange order.
Decision / selection error
Why it happens
Once curves cross, the previously top curve may become the bottom curve on the next interval.
How it is corrected
Re-test curve order with a sample point after every intersection before integrating the next piece.
Splitting at irrelevant points but missing sign-changing points.
Execution error
Why it happens
Only points where sign or curve order actually changes require a split; other points do not.
How it is corrected
Identify sign-changing and order-changing points specifically, rather than splitting arbitrarily.
Forgetting that an odd-function integral on a symmetric interval can be zero while geometric area is positive.
Knowledge gap
Why it happens
A zero signed integral from odd symmetry does not imply the bounded geometric area is zero.
How it is corrected
Use |f(x)|, or split at the zero, to compute geometric area rather than relying on the signed symmetric result.
Calling every definite integral an area problem.
Decision / selection error
Why it happens
A definite integral is a signed accumulation; it becomes a geometric-area problem only when the question asks for area explicitly.
How it is corrected
Check whether the question asks for the signed integral value or for geometric area before answering.

FAQ