Area Under Curve Formula Definition and Solved Examples
Calculus is the mathematical study of change in the ways of geometrical shapes and algebra for the simplification of arithmetic calculation. Ideally, there are two major branches of calculus, namely integral calculus and differential calculus.
There are two major branches of calculus, namely integral calculus and differential calculus. Here integral calculus is the gathering of quantities, and areas between or area under curve calculus. While differential calculus concerns instant rates of change and the slopes of curves.
Students will learn how these two branches are interlinked to form fundamental theorems of calculus under this section, as they use the basic concept of convergence of infinite series and sequences to a defined limit. Here the formula to calculate area between two curves calculus is explained in detail.
Ways of Finding the Area under a Curve Calculus
Here is an example to show how the area under curve calculus is calculated. Take a look!
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In the above figure, one can assume the curve y=f(x) and its ordinates of the x-axis to be x=a and x=b. To understand how to find the area under the curve, students have to assess the area surrounded by the given curve and its ordinates show that x=a and x=b.
Here one can find the definite integral area under the curve. Taking a random strip of height as Y and width as dx. In the figure above, dA is assumed to be the area.
Now the dA area of the strip is provided with a dx while a point in the curve that is y is represented through f(x). The strip areas between curves calculus can also be termed as an elementary area between the x-axis. And the curve is located between x=a and x=b. To find the total is bounded by this curve, one has to consider there is an infinite number of strips. These strips start from x=b to x=a. Adding an elementary area between given strips in the region PQRSP helps find the total area needed.
How to Find an Integral Calculus Area under a Curve Mathematically?
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To calculate the area of the curve by integration or under the curve, let’s take x=g(y). Here between the lines y=c and y=d lies y-axis. This can be given by the formula A=∫cdxdy=∫cdg(y)dy.
One has to keep in mind that there is a horizontal strip like the image above. If a curve lies underneath the x-axis where f(x) is less than zero then similar steps will give the needed area under the curve. This will be negative in value as it lies under x-axis and between x=a and x=b.
It is suggested to take an absolute value of an area without the .|∫abf(x)dx| sign to find the area under curve calculus.
To determine the calculus 2 area between curves, one has to take an example where the area between y=g(x) y=g(x) and y=f(x) y=f(x) is interval [a,b]. Let’s take an assumption that f(x) is less than equal to g(x).
To find the exact value, we have to use the formula A=∫dcf(y)−g(y)dy. It is vital to remember that the first function value is larger. It is smarter to use the term formulas to show that area will be larger minus smaller function.
Apart from these formulas, students need to practice calculation of area bounded by a curve and a line for a better score in boards. Every student desires quality practice material to excel in competitive exams.
They can check Vedantu, which is a pocket-friendly e-learning portal offering area between curves practice questions and more. They also provide live classes, top-notch notes on the integration of area between two curves with detailed examples. To enjoy these features check the official site today
FAQs on Area Under Curve in Calculus Explained Clearly
1. What is the area under a curve in calculus?
The area under a curve is the region between a function’s graph and the x-axis over a given interval, calculated using a definite integral. In calculus, it represents the accumulation of quantities such as distance, total change, or probability.
- It is written as ∫ab f(x) dx.
- If f(x) ≥ 0 on [a, b], the integral equals the actual area.
- If f(x) is below the x-axis, the integral gives a negative value.
2. What is the formula for finding the area under a curve?
The formula for the area under a curve from x = a to x = b is ∫ab f(x) dx. This is called the definite integral.
- First, find the antiderivative F(x) of f(x).
- Then apply the Fundamental Theorem of Calculus: F(b) − F(a).
- Antiderivative: F(x) = x³/3
- Area = (8/3 − 0) = 8/3
3. How do you calculate the area under a curve step by step?
To calculate the area under a curve, evaluate the definite integral using an antiderivative. Follow these steps:
- Step 1: Write the integral ∫ab f(x) dx.
- Step 2: Find the antiderivative F(x).
- Step 3: Substitute limits: compute F(b) − F(a).
- Antiderivative: (3x²)/2
- Area = (27/2 − 3/2) = 12
4. What does a negative area under a curve mean?
A negative area under a curve means the function lies below the x-axis, so the definite integral value is negative. In definite integration:
- If f(x) > 0, area is positive.
- If f(x) < 0, integral value is negative.
5. What is the difference between definite and indefinite integrals?
A definite integral gives a numerical area over an interval, while an indefinite integral gives a family of antiderivatives. Key differences:
- Definite: ∫ab f(x) dx → single number.
- Indefinite: ∫ f(x) dx = F(x) + C → includes constant C.
- Definite integrals are used for area and accumulation.
6. How is the area under a curve related to the Fundamental Theorem of Calculus?
The Fundamental Theorem of Calculus states that a definite integral equals the difference of an antiderivative at the endpoints. It connects differentiation and integration:
- If F′(x) = f(x), then ∫ab f(x) dx = F(b) − F(a).
- This makes area calculation systematic and exact.
7. Can you give an example of finding the area under a curve?
Yes, for f(x) = 2x + 1 from 0 to 2, the area under the curve is 6. Solution:
- Antiderivative: F(x) = x² + x
- Evaluate: F(2) − F(0)
- = (4 + 2) − 0 = 6
8. How do you find the area between two curves?
The area between two curves is found using ∫ab [f(x) − g(x)] dx, where f(x) is above g(x). Steps:
- Find intersection points to determine limits.
- Subtract lower function from upper function.
- Evaluate the definite integral.
9. Why do we use integration to find area under a curve?
We use integration because it sums infinitely many infinitesimal rectangles to give the exact area. The process is based on limits:
- Divide region into thin strips of width dx.
- Area of each strip ≈ f(x) dx.
- Sum using the integral symbol ∫.
10. What are common mistakes when finding area under a curve?
Common mistakes include forgetting limits, sign errors, and incorrect antiderivatives. Watch out for:
- Missing the constant in indefinite integrals.
- Not applying F(b) − F(a) correctly.
- Ignoring negative areas below the x-axis.
- Using wrong upper and lower functions between curves.
