medium difficulty10 min

Area Bounded by Standard Curves

Area bounded by standard curves is found by first sketching the known curve shape, identifying the required bounded region, and then setting a definite integral with correct limits. Common curves include circles, parabolas, ellipses and straight lines.

In one line: Standard curve first, shaded part second, integral third.

Practice This Concept

Understand it

Standard curves often require choosing the correct branch or using symmetry.
For a circle x² + y² = a², the upper half is y = √(a² - x²).
For a parabola y² = 4ax, horizontal strips may be easier because x can be written in terms of y.
The diagram is not optional in reasoning; it decides which expression, limits, and multiplier are valid.

Example

The area of the upper semicircle x² + y² = a² is ∫_-a^a √(a² - x²) dx = πa²/2 square units.

Formula and rule

Useful forms and conditions: Circle x² + y² = a² has upper branch y = √(a² - x²), -a ≤ x ≤ a.

Parabola y² = 4ax gives x = y²/(4a), useful with dy.

Ellipse x²/a² + y²/b² = 1 has upper branch y = b√(1 - x²/a²).

Use symmetry only when the shaded region is symmetric about an axis.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Find the area enclosed by the ellipse x²/16 + y²/9 = 1.

1The ellipse is symmetric about both axes.
2In the first quadrant, y = 3√(1 - x²/16), with x from 0 to 4.
3Total area = 4∫₀⁴ 3√(1 - x²/16) dx.
4This equals 4 times the first-quadrant area of an ellipse with semi-axes 4 and 3, so area = πab = π(4)(3) = 12π square units.
5The integral setup and symmetry justify the result.

Method you can reuse

1Recognise the standard curve and note its intercepts or vertex.
2Draw the bounded region.
3Decide whether vertical or horizontal strips give a simpler expression.
4Convert the curve equation into y = f(x) or x = g(y).
5Set limits from intercepts or intersection points.
6Apply symmetry only if the shaded region repeats exactly.
7Integrate or use a justified standard area result when allowed.

Sketch it in your answer

Sketch the standard curve accurately enough to show intercepts, axes of symmetry, and the bounded part. Label intercepts such as (-a, 0), (a, 0), (0, b), and mark whether the upper, lower, left or right branch is used.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

For the parabola y² = 4ax, which strip direction is usually convenient for area between y = 0 and y = b?

Think first, then show answer

Horizontal strips are convenient because x = y²/(4a), so the area can be written using dy limits from 0 to b.

Common wrong answer

For x²/16 + y²/9 = 1, writing the area as π(16)(9) instead of π(4)(3) confuses denominators with semi-axis lengths.

Don't mix this up

A frequent error is using the full circle or full ellipse area formula when the question asks for only one bounded part.

Exam tip

For standard curves, write the curve in the form needed by the strip direction. Use vertical strips for y as a function of x and horizontal strips for x as a function of y.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Questions commonly ask for area enclosed by a circle, ellipse, parabola with a line, or a region in one quadrant. Marks are awarded for a correct sketch, correct limits, justified symmetry, and valid integral setup.

1
Enough for 1 mark
Area bounded by standard curves is found by first sketching the known curve shape, identifying the required bounded region, and then setting a definite integral with correct limits. Common curves include circles, parabolas, ellipses and straight lines.
2
Add for 2 marks
Example: The area of the upper semicircle x² + y² = a² is ∫_-a^a √(a² - x²) dx = πa²/2 square units.
3
Add for 3 marks
Standard curves often require choosing the correct branch or using symmetry.

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