How to Evaluate Definite Integrals Using Formula and Limits
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 with Step by Step Solutions
1. What is a definite integral?
A definite integral is the value of an integral evaluated between two limits that represents the net area under a curve. It is written as ∫ab f(x) dx, where a and b are the lower and upper limits.
It gives:
- The signed area between the curve and the x-axis.
- A numerical result (not a function).
- The accumulation of quantities such as area, displacement, or total change.
2. How do you calculate a definite integral?
To calculate a definite integral, find the antiderivative and apply the limits using the Fundamental Theorem of Calculus. The formula is:
∫ab f(x) dx = F(b) − F(a), where F'(x) = f(x).
Steps:
- Find the antiderivative F(x).
- Substitute the upper limit b.
- Substitute the lower limit a.
- Subtract: F(b) − F(a).
3. What is the formula for a definite integral?
The main formula for a definite integral is ∫ab f(x) dx = F(b) − F(a). This comes from the Fundamental Theorem of Calculus.
Where:
- f(x) is the integrand.
- F(x) is the antiderivative of f(x).
- a and b are the limits of integration.
4. What does a definite integral represent?
A definite integral represents the net signed area under a curve between two points. Specifically, ∫ab f(x) dx measures:
- Area above the x-axis as positive.
- Area below the x-axis as negative.
- Total accumulated change of a quantity.
5. What is the difference between definite and indefinite integrals?
The key difference is that a definite integral gives a numerical value, while an indefinite integral gives a function plus a constant.
- Definite integral: ∫ab f(x) dx = number.
- Indefinite integral: ∫ f(x) dx = F(x) + C.
6. How do you use a definite integral calculator?
To use a definite integral calculator, enter the function and specify the lower and upper limits correctly. Follow these steps:
- Input the integrand, such as x^2 or sin(x).
- Enter the lower limit (a).
- Enter the upper limit (b).
- Click calculate to get the result.
7. Can you give an example of solving a definite integral?
Yes, for example, ∫04 3x² dx = 64.
Solution:
- Antiderivative of 3x² is x³.
- Evaluate: [x³]04.
- Compute: 4³ − 0³ = 64 − 0.
8. What is the Fundamental Theorem of Calculus?
The Fundamental Theorem of Calculus states that integration and differentiation are inverse processes. It says:
∫ab f(x) dx = F(b) − F(a), where F'(x) = f(x).
This theorem allows you to evaluate definite integrals using antiderivatives instead of limits of sums.
9. What happens if you reverse the limits of a definite integral?
If you reverse the limits of a definite integral, the sign of the result changes. The rule is:
∫ab f(x) dx = −∫ba f(x) dx.
This happens because the direction of accumulation is reversed.
10. Why is my definite integral calculator giving a negative answer?
A definite integral calculator gives a negative answer when the function is mostly below the x-axis over the interval. Since definite integrals measure signed area:
- Area above the x-axis is positive.
- Area below the x-axis is negative.
