What is Definite Integral Solver?
The subject of mathematics is incredibly diverse. There are near-infinite concepts and topics, and one of them is integral. We can understand an integral as either a numerical value equal to the area under the graph of a function for some interval, known as the definite integral. We can integrate a definite integral of any function by using the indefinite integral and a corollary to the fundamental theorem of calculus. Before getting into the nuances of the definite integral calculator, let us analyze what integration is.
The definition of integration is related to the application of definite integrals as well indefinite integrals. In simple words, integration is the calculation of an integral. As we have already discussed, an integral is used to find many valuable quantities such as areas, volumes, displacement, and so forth. It is noteworthy to illustrate that when we say integra, we usually mean definite integrals. As such, the definite integral solver becomes crucial in Calculus and mathematics.
Integration represents the sum of discrete data. It is calculated to extract the functions that will describe aspects such as area, displacement, volume, and so forth that occur due to the amalgamation of small data that we cannot measure singularly. Now that we have touched upon the fundamentals of integration, let us try and explore the nitty-gritty of integral calculus.
A Brief Discussion on Integral Calculus
As per pioneering mathematician Bernhard Riemann – an integral is premised on a limiting or restricting procedure that assumes the area of a curvilinear region by breaking down the part into thin slabs of vertical length. For example, we can investigate the slope by using a slope formula, but if we want to find an area of a curve, we have to be equipped with differential calculus. You can use the Wolfram integral calculator to grasp the topic of integral and integration further.
Definite Integral Calculator – Standard Format and Formula
The definite integral is also deemed as the Riemann integral. The definite integral of a function is equal to the area of the region bounded by the curve. A definite integral relates to numerical values and is used in pure mathematics, applied mathematics, statistics, science, and so forth. To get into the crux of the definite integral solver, we have to know the formula of definite integrals which are –
\[\int_{b}^{a}f(x)dx\] = \[-\int_{a}^{b}f(x)dx\]
\[\int_{b}^{a}f(x)dx\] = 0
\[\int_{a}^{b}cf(x)dx\] = \[c\int_{a}^{b}f(x)dx\]
\[\int_{a}^{b}f(x)\pm g(x)dx\] = \[\int_{a}^{b}f(x)dx\] $\pm$ \[\int_{a}^{b}g(x)dx\]
\[\int_{a}^{b}f(x)dx\] = \[\int_{a}^{c}f(x)dx\] + \[\int_{c}^{b}f(x)dx\]
Wolfram Definite Calculator – Trigonometric Functions
Wolfram Alpha is a computational knowledge engine and answer engine developed by Wolfram research. It answers questions related to definite integers, trigonometry, and so forth. It is an important tool for people who want answers to complex questions. As such, we can understand the definite integral calculator with steps.
Applications of Definite Integral
The use of definite integrals is manifold and diverse. They are used in probability theory to determine the probability of some random variable falling within a specific range. Integrals can also compute the area of a two-dimensional region with a curved boundary and calculate the volume of three-dimensional objects with a curved perimeter. Integrals are also utilized in physics in kinematics, displacement, time, velocity, and so forth. They also have a crucial role in thermodynamics. You can opt for a wolfram integral calculator to know more about the topic.
Definite Integral Solver to Find the Area of a Circle
It is noteworthy to mention that definite integrals can be easily used to calculate the area of a circle.
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The area of the circle is extracted by calculating the area of the part of the circle in the first quadrant, and the equation used is x²+y²=a². We will find the area of the one-quarter circle and multiply it by 4 to determine the circle's total area since circles are symmetric around the x and y axes.
Find the value of y in the given equation above.
After that, the equation is changed to the equation of a curve \[Y=\sqrt{a^2-x^2}\].
Then, we utilize the concept of definite integral to extract the equation of the curve and give the limits from '0' to 'a'. The area of a circle is four times the area of a quadrant. The area of the quadrant is calculated by integrating the equation of the curve across the limits in the first quadrant.
∴ Area of circle =4 × Area of first quadrant
\[A=\int_{0}^{a}y.dx\]
Here, putting the value of y from above, we get,
= \[4\int_{0}^{a}\sqrt{a^2-x^2}.dx\]
= \[4[\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}sin^{-1}\frac{x}{a}]_0^a\]
= \[4[0+\frac{a^2}{2}sin^{-1}\frac{a}{a}-(0+\frac{a^2}{2}sin^{-1}\frac{0}{a})]\]
= \[4[\frac{a^2}{2}sin^{-1}1-0]\]
= \[4(\frac{a^2}{2})(\frac{\pi }{2})\]
= \[\pi a^2\]
Thus, the area of the circle is \[\pi a^2\] square units.
Area of a Parabola by Using a Definite Integral
We know that a parabola comprises an axis that divides it into two equal or symmetric parts.
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A parabola that is equal along the x-axis and has an equation of y²=4ax. Then we transform the equation into y=√4ax. We begin by finding the area of the parabola in the first quadrant by utilizing the definite integral formulas vis-à-vis the x-axis and the limits from '0' to 'a'. After that, the definite integral can be extracted within the boundary, and we have to double it to extract the area of the whole parabola.
\[A=2\int_{0}^{a}\sqrt{4ax}.dx\]
= \[4\sqrt{a}\int_{0}^{a}\sqrt{x}.dx\]
= \[4\sqrt{a}[\frac{2}{3}.x^\frac{3}{2}]_0^a\]
= \[4\sqrt{a} (\frac{2}{3}a^\frac{3}{2})-0\]
= \[\frac{8a^2}{3}\]
Hence, the area under the curve covered by the parabola is \[\frac{8a^2}{3}\] square units.
Area of An Ellipse By Using a Definite Integral
We know that the ellipse equation with the major axis of length 2a and a minor axis of 2b is \[\frac{x^2}{a^2} + \frac{y^2}{b^2}=1\] . Then we have to transform the equation to \[y=\frac{b}{a}\sqrt{(a^2-x^2)}\] .
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After that, the definite integral concept has to be appropriated to calculate the area bounded by the ellipse in the first coordinate and the x-axis. Then, we have to multiply the equation by 4 to obtain the area of the ellipse.
\[A=4\int_{0}^{a}y.dx\]
= \[4\int_{0}^{a}\frac{b}{a}(a^2-x^2).dx\]
= \[4\frac{b}{a}[\frac{x}{2}.\sqrt{a^2-x^2} + \frac{a^2}{2}sin^{-1}\frac{x}{a}]_0^a\]
= \[4\frac{b}{a}(\frac{a}{2}\times0)+\frac{a^2}{2}sin^{-1}-0]\]
= \[4\frac{b}{a}\times\frac{a^2}{2}\times\frac{\pi }{2}\]
= \[\pi ab\]
Thus, the area of the ellipse is \[\pi ab\] sq units.
In conclusion, definite integral is a crucial part of mathematics and forms the basis of various branches of the discipline. It is imperative that we know the basics of the topic.
FAQs on Definite Integral Calculator
1. What is a definite integral and what does it represent geometrically?
A definite integral, denoted as ∫ab f(x)dx, represents the accumulated value of a function f(x) over a specific interval [a, b]. Geometrically, it calculates the net area of the region bounded by the graph of the function y = f(x), the x-axis, and the vertical lines x = a (the lower limit) and x = b (the upper limit). If the function is above the x-axis, the integral gives a positive area; if it's below, it contributes a negative area.
2. How is a definite integral calculated using the Fundamental Theorem of Calculus?
According to the Fundamental Theorem of Calculus, if F(x) is the antiderivative (indefinite integral) of a continuous function f(x) on the interval [a, b], then the definite integral of f(x) from a to b is calculated by finding the difference in the value of the antiderivative at the upper and lower limits. The formula is: ∫abf(x)dx = F(b) - F(a). This provides a direct method for evaluation without using the limit of a sum.
3. What are the most important properties of definite integrals?
Definite integrals have several key properties that simplify calculations, as per the CBSE/NCERT syllabus for the 2025-26 session. Some of the most important ones include:
- Order of Integration: ∫abf(x)dx = -∫baf(x)dx
- Zero Interval: If a = b, then ∫aaf(x)dx = 0
- Constant Multiple: ∫abc·f(x)dx = c·∫abf(x)dx, where c is a constant.
- Sum/Difference Rule: ∫ab[f(x) ± g(x)]dx = ∫abf(x)dx ± ∫abg(x)dx
- Additivity of Interval: ∫abf(x)dx = ∫acf(x)dx + ∫cbf(x)dx, where a < c < b.
4. What are some real-world applications of definite integrals beyond finding area?
While finding the area under a curve is a primary application, definite integrals are crucial in many other fields. For example:
- In Physics, they are used to calculate total displacement from a velocity function, work done by a variable force, or the centre of mass of an object.
- In Economics, they can determine the total revenue from a marginal revenue function or find the consumer and producer surplus.
- In Probability and Statistics, definite integrals are used to find the probability of a continuous random variable falling within a certain range.
- In Engineering, they help calculate the volume of solids of revolution and the pressure exerted by a fluid.
5. How does an online definite integral calculator work?
An online definite integral calculator typically uses a two-step process based on the Fundamental Theorem of Calculus. First, it performs symbolic integration to find the antiderivative (the indefinite integral) of the given function. Second, it evaluates this antiderivative at the upper and lower limits of integration and subtracts the results. For functions that cannot be integrated symbolically, the calculator uses numerical approximation methods (like the Trapezoidal Rule or Simpson's Rule) to estimate the area under the curve.
6. Why can a definite integral have a negative value, and what does it signify?
A definite integral can be negative. This happens when the area of the region bounded by the curve below the x-axis is greater than the area of the region bounded by the curve above the x-axis within the given interval. A negative result for a definite integral signifies that the net accumulation is negative. For instance, if the integral represents displacement and the result is negative, it means the final position is behind the starting position.
7. What is the fundamental difference between a definite integral and an indefinite integral?
The core difference lies in what they represent. An indefinite integral (or antiderivative) of a function f(x) is a family of functions, written as F(x) + C, where C is the constant of integration. In contrast, a definite integral, ∫abf(x)dx, is a single numerical value. It represents the net accumulation or area over a specified interval [a, b] and has no constant of integration.
8. In what situations might you need to approximate a definite integral instead of calculating it exactly?
Exact calculation of a definite integral is not always possible. Approximation is necessary in two main scenarios:
- When the function, f(x), does not have an elementary antiderivative. A classic example is the function e-x², which is crucial in statistics but cannot be integrated using standard techniques.
- When the function is not defined by a formula but by a set of discrete data points or a graph obtained from an experiment. In such cases, numerical methods are the only way to estimate the integral.
