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294 LearnersLast updated on September 17, 2025
Area of Parabola
What is the Area Under a Parabola?
A parabola is a U-shaped curve that can open upwards or downwards. It is the graph of a quadratic function of the form y = ax² + bx + c.
The area under a parabola between two points on the x-axis can be calculated using definite integration. This area represents the space enclosed between the parabola and the x-axis.
Formula for the Area Under a Parabola
To find the area under a parabola from x = a to x = b, we use definite integration. The formula is: Area = ∫[a to b] (ax² + bx + c) dx Let’s see how the formula is derived.
Derivation of the formula: 1. Start with the quadratic function y = ax² + bx + c. 2. Integrate the function with respect to x: ∫(ax² + bx + c) dx = (a/3)x³ + (b/2)x² + cx + C, where C is the constant of integration. 3. Evaluate the definite integral from x = a to x = b: Area = [(a/3)b³ + (b/2)b² + cb] - [(a/3)a³ + (b/2)a² + ca]
How to Find the Area Under a Parabola?
We can find the area under a parabola using definite integration. Here’s a step-by-step method:
1. Identify the quadratic function y = ax² + bx + c.
2. Determine the interval [a, b] over which you want to find the area.
3. Set up the integral: ∫[a to b] (ax² + bx + c) dx.
4. Integrate the function and evaluate the definite integral over [a, b].
For example, if y = 2x² + 3x + 1 and you want to find the area from x = 0 to x = 2: Area = ∫[0 to 2] (2x² + 3x + 1) dx
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Units of the Area Under a Parabola
The area under a parabola is measured in square units. The unit depends on the measurement system used for the x and y axes:
In the metric system, it might be square meters (m²), square centimeters (cm²), etc.
In the imperial system, it could be square inches (in²), square feet (ft²), etc.
Special Cases or Variations for the Area Under a Parabola
Depending on the form of the quadratic function and the interval, the area calculation can vary. Here are some special cases:
Case 1: Parabola symmetric about the y-axis If the parabola is symmetric about the y-axis, such as y = ax², integrate over [-b, b].
Case 2: Parabola intersecting x-axis If the parabola intersects the x-axis at points x = a and x = b, find the area between these points.
Case 3: Applications in physics In physics, finding the area under a velocity-time graph (a parabola) can represent the displacement of an object.
Tips and Tricks for Calculating the Area Under a Parabola
To ensure accurate results while calculating the area under a parabola, consider these tips:
- Verify the limits of integration to ensure they correspond to the correct interval.
- Use symmetry properties of the parabola to simplify calculations if applicable.
- Double-check the integration process, especially the substitution of limits.
Common Mistakes and How to Avoid Them in Calculating Area Under a Parabola
Errors can occur when calculating the area under a parabola. Let’s examine some common mistakes.
Mistake 1
Incorrect integration of the quadratic function
Ensure the integration of the quadratic function is performed correctly.
Missteps in the integration process can lead to incorrect area calculations.
Mistake 2
Using wrong limits of integration
Check that the limits of integration correspond to the correct interval on the x-axis.
Incorrect limits can result in an inaccurate area.
Mistake 3
Confusing the parabola with other curves
Ensure you are working with a quadratic function and not another type of curve, such as a circle or cubic function.
Mistake 4
Forgetting to evaluate definite integral
After integrating, remember to evaluate the definite integral over the specified interval to find the exact area.
Mistake 5
Omitting units from the final answer
Always include the correct units in your final answer to convey the area accurately.
Area Under a Parabola Examples
Problem 1
Calculate the area under the parabola y = x² + 2x + 3 from x = 1 to x = 4.
We will find the area as 39 square units.
Explanation
The quadratic function is y = x² + 2x + 3, and the interval is [1, 4].
Area = ∫[1 to 4] (x² + 2x + 3) dx = [(1/3)x³ + x² + 3x] (from 1 to 4) = [(1/3)(4)³ + (4)² + 3(4)] - [(1/3)(1)³ + (1)² + 3(1)] = [64/3 + 16 + 12] - [1/3 + 1 + 3] = [92/3 + 28] - [4.333] = 39 square units
Problem 2
Find the area under the parabola y = 3x² - 2x + 1 from x = 0 to x = 3.
We will find the area as 24.5 square units.
Explanation
The quadratic function is y = 3x² - 2x + 1, and the interval is [0, 3].
Area = ∫[0 to 3] (3x² - 2x + 1) dx = [(x³) - x² + x] (from 0 to 3) = [(3³) - 3² + 3] - [0] = [27 - 9 + 3] = 21 square units
Problem 3
The area under the parabola y = 4x² - x + 2 from x = -1 to x = 2 is given as 14 square units. Verify this area.
We find the calculated area as 14 square units.
Explanation
The quadratic function is y = 4x² - x + 2, and the interval is [-1, 2].
Area = ∫[-1 to 2] (4x² - x + 2) dx = [(4/3)x³ - (1/2)x² + 2x] (from -1 to 2) = [(4/3)(2)³ - (1/2)(2)² + 2(2)] - [(4/3)(-1)³ - (1/2)(-1)² + 2(-1)] = [32/3 - 2 + 4] - [-4/3 - 1/2 - 2] = 14 square units
Problem 4
If the parabola y = 5x² + x - 1 is given, find the area from x = -2 to x = 1.
We will find the area as 26.5 square units.
Explanation
The quadratic function is y = 5x² + x - 1, and the interval is [-2, 1].
Area = ∫[-2 to 1] (5x² + x - 1) dx = [(5/3)x³ + (1/2)x² - x] (from -2 to 1) = [(5/3)(1)³ + (1/2)(1)² - (1)] - [(5/3)(-2)³ + (1/2)(-2)² - (-2)] = [5/3 + 1/2 - 1] - [-40/3 + 2 - 2] = 26.5 square units
Problem 5
Help Emma find the area under the parabola y = 2x² - 3x + 4 from x = 0 to x = 5.
We will find the area as 95 square units.
Explanation
The quadratic function is y = 2x² - 3x + 4, and the interval is [0, 5].
Area = ∫[0 to 5] (2x² - 3x + 4) dx = [(2/3)x³ - (3/2)x² + 4x] (from 0 to 5) = [(2/3)(5)³ - (3/2)(5)² + 4(5)] - [0] = [250/3 - 75/2 + 20] = 95 square units
FAQs on Area Under a Parabola
1.Is it possible for the area under a parabola to be negative?
2.How to find the area under a parabola if the parabola is below the x-axis?
3.What if the parabola opens downwards?
4.How do you calculate the area if the parabola is symmetric?
5.What is meant by the area under a parabola?
Glossaries of Terms Related to Area Under a Parabola
- Area: The total space occupied by a shape, measured in square units.
- Parabola: A U-shaped curve that is the graph of a quadratic function.
- Definite Integration: A calculus technique to calculate the area under a curve between two limits.
- Symmetry: A property where a shape is identical on both sides of a dividing line.
- Quadratic Function: A polynomial function of degree two, generally written as y = ax² + bx + c.
Seyed Ali Fathima S
About the Author
Seyed Ali Fathima S a math expert with nearly 5 years of experience as a math teacher. From an engineer to a math teacher, shows her passion for math and teaching. She is a calculator queen, who loves tables and she turns tables to puzzles and songs.
Fun Fact
: She has songs for each table which helps her to remember the tables
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