Fundamental theorem of calculus

Introduction

The fundamental theorem of Calculus is a crucial theorem in Calculus that connects antiderivatives with definite integrals. The fundamental theorem of calculus asserts that if a function f has an antiderivative F, then the definite integral of f from a to b equals F(b)-F(a). This theorem is useful for determining a function’s net change, area, or average value over a region.

Theorem of Calculus

Calculus is a field of Mathematics that aids in the understanding of changes in values that are connected to a function. Physics, Engineering, Medicine, Economics, Biology, Space Exploration, Statistics, Pharmacology, and many more fields use it. Even a house cannot be built without calculus.

Calculus has a long history dating back to Ancient Egypt. Integral calculus, according to historians, was employed by Egyptians to compute the volume of a pyramidal frustum. They were familiar with integral calculus’ basic functions and could apply them to determine volumes and areas. Modern calculus, on the other hand, was invented separately by Isaac Newton and Gottfried Wilhelm Leibniz.

Calculus Fundamental Theorem

The powerful theorem in mathematics is the fundamental theorem of calculus. It established a differentiation-integration relationship. This relationship now allows us to evaluate definite integrals without having to calculate areas or use Riemann sums. There are two parts to the fundamental theorem:

• The first fundamental theorem
• The second fundamental theorem

The Fundamental Theorem of Calculus says that differentiation and integration are inverse operations. In other words, they undo each other.

To make an analogy: This is like multiplication and division—they are inverse operations and undo each other. Take 8 and divide it by 2, and you get 4. Multiply the answer by 2, which is 2 times 4, and you get what you started with, 8. Multiplication by 2 undoes division by 2. Differentiation and integration undo each other in the same way.

Here’s another example of how differentiation and integration undo each other. Start with the function f(X) = 2X. Integrate f(X) = 2X and you get the function f(X) = X squared. Differentiate this function, f(X) = X squared, and you get back the function f(X) = 2X.

Graphing these functions may give an intuitive sense of what is going on. f(X)=2X graphs as a straight line. If you integrate this function, you’re finding the area under the curve, that is, the area shown in yellow in the accompanying image.

Graph the amount of area under the curve as X increases, and you’ll be graphing the integral f(X) =X squared, a parabola. Graph the function which tells you the slope of the tangent to every point on the parabola, and you’ll be graphing its derivative, the straight line, f(X)=2X.

Integral Calculus

Integral calculus is used to find a function’s antiderivatives. The function’s integrals are another name for these antiderivatives. Integration is the process of calculating the antiderivative of a function.

The procedure of determining integrals is the inverse of that of determining derivatives. The integral of a function represents a family of curves. Finding derivatives and integrals is a requirement of basic calculus. In what follows, we’ll go over the principles of integrals and how to evaluate them.

Integral Methodologies

The indefinite integrals can be found using a variety of ways. The following are some of the most popular methods:

• Integrals can be found using the integration by substitution approach.
• Integrals are discovered through part-by-part integration.
• Integrals can be found by integrating partial fractions.

Integral Calculus’s Characteristics

Let’s look at the properties of indefinite integrals to work with them.

• An integral’s derivative is called an integrand.

f(x) dx = f(x) +C f(x) dx = f(x) dx = f(x) dx

• Two indefinite integrals are comparable as they produce the same derivative with the same family of curves.

0 = [f(x) dx -g(x) dx]

• The sum or the difference of functions that are finite in nature and that of the individual functions’ integrals are equal.

f(x) dx + g(x) dx = [f(x) dx+g(x) dx] = f(x) dx + g(x) dx

• It is necessary to take the constant outside the integral sign.

k f(x) dx = k f(x) dx = k f(x) dx, where k R.

Conclusion

The fundamental theorem of Calculus is a crucial theorem in Calculus that connects antiderivatives with definite integrals. The fundamental theorem of calculus asserts that if a function f has an antiderivative F, then the definite integral of f from a to b equals F(b)-F(a).

Calculus has a long history dating back to ancient Egypt. The calculus fundamental theorem is a powerful theorem in Mathematics and has numerous practical applications.