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 Calculus
1. What is the fundamental concept behind finding the area under a curve using calculus?
The core idea is to approximate the area under a curve by dividing it into an infinite number of extremely thin vertical rectangles. Each rectangle has a height of f(x) (the function's value) and an infinitesimally small width, denoted as dx. Calculus, specifically definite integration, provides a powerful method to sum the areas of all these infinite rectangles to find the exact total area under the curve between two points.
2. What is the primary formula used to calculate the area under a curve y = f(x)?
The area (A) bounded by the curve y = f(x), the x-axis, and the vertical lines x = a and x = b is calculated using the definite integral formula:
A = ∫ₐᵇ y dx = ∫ₐᵇ f(x) dx
Here, 'a' is the lower limit of integration (the starting x-coordinate) and 'b' is the upper limit (the ending x-coordinate). The integral of f(x) gives the antiderivative, which is then evaluated at the limits b and a.
3. How do you calculate the area under a curve that is defined with respect to the y-axis, like x = g(y)?
When a curve is defined as a function of y, we change our perspective. Instead of vertical rectangles, we use an infinite number of thin horizontal rectangles. The area (A) bounded by the curve x = g(y), the y-axis, and the horizontal lines y = c and y = d is given by the formula:
A = ∫ₐᵇ x dy = ∫ₐᵇ g(y) dy
Here, we integrate the function with respect to 'y' from the lower limit 'c' to the upper limit 'd'.
4. What happens if a curve lies below the x-axis? Does the area become negative?
Yes, if the curve y = f(x) is below the x-axis over an interval [a, b], the definite integral ∫ₐᵇ f(x) dx will yield a negative value. This is because the function values f(x) are negative in that region. However, since geometric area cannot be negative, we take the absolute value of the integral to find the actual area. The required area is A = |∫ₐᵇ f(x) dx|.
5. How is the method for finding the area between two intersecting curves different from finding the area under a single curve?
To find the area between two curves, y = f(x) and y = g(x), where f(x) ≥ g(x) over an interval [a, b], you essentially find the area under the upper curve and subtract the area under the lower curve. The consolidated formula is:
A = ∫ₐᵇ [f(x) - g(x)] dx
This calculates the area of the region enclosed directly between the two functions, rather than between one function and an axis.
6. According to the CBSE Class 12 syllabus for 2025-26, what are the typical shapes for which students need to calculate the area?
As per the NCERT and CBSE curriculum, students should be proficient in finding the area under simple curves, especially:
- Lines: Linear equations like y = mx + c.
- Circles: Equations of the form x² + y² = r².
- Parabolas: Standard forms like y² = 4ax and x² = 4ay.
- Ellipses: Standard form x²/a² + y²/b² = 1.
Problems often involve finding the area of regions bounded by a combination of these curves and lines.
7. Why is the concept of 'area under a curve' important in real-world applications beyond mathematics?
This concept has significant applications in various fields. For example:
- In Physics, the area under a velocity-time graph gives the total displacement, and the area under a force-displacement graph represents the work done.
- In Economics, it is used to calculate concepts like consumer surplus and producer surplus from supply and demand curves.
- In Statistics, the area under a probability density function curve represents the probability of an outcome falling within a certain range.
8. Is it possible to find the area under a curve without using integration?
While integration provides the exact area, it is possible to approximate the area without it. Methods like the Riemann Sum (using a finite number of rectangles) or the Trapezoidal Rule can give a close estimate. However, these are approximation techniques. Calculus, through definite integrals, is the tool used to sum an infinite number of infinitesimally small parts, thereby moving from an approximation to an exact value.
