How to learn integration in school?

Integrals are the basic part of Calculus formulated by Isaac Newton and Gottfried Wilhelm Leibniz. The basic definition of integrals explains it as the area under the curve as an infinite sum of rectangles of minuscule width. Integrals are the inverse of derivatives so are called anti-derivative. The process of finding integrals is called integration. The symbol used for integration is “⌠.” Integral calculator helps us to determine the integrals.

Derivation of integrals:

suppose we wish to determine the area under the function. Divide the region into four segments. And to draw a rectangle inside each segment such that the upper right corner of the rectangle touches the function. The sum of the area of rectangles is an estimated area under the function. The estimate is poor in this case because the rectangles do not fit under the curve. It is because they are too wide as shown in the figure below.

Now if we want to find the exact area under the curve of a specific function we have to limit the width of each rectangle. We can do this by limiting their width 6 times. As it is shown in the graph below.

These narrower rectangles fit under the curve better than the wider ones. Now the sum of the area is a much better estimate of the area under the curve. You can do this differently by placing rectangles into four segments such the left edge of each rectangle touches the curve.

Riemann Sum and Integrals:

Now we are going to translate the whole concept into a mathematical explanation. Let’s imagine we want to find the area under the same curve between these two x-coordinates. We will estimate the area inside three equally sized regions to translate it into maths we need d labels for x-coordinates. X-coordinates labels define the range of each of these regions from left to right. Let the coordinates be x₀, x₁, x₂, x₃. As the figure shows.

Now place a rectangle between x₀ and x₁ such that the upper right corner touches the curve x₁. The area of this rectangle is its height which is the value of the function at x₁ multiplied by its width which is ( x₁ – x₀). In this way add the area of all three rectangles that exactly fit in the upper shown graph. Let’s call the area of all three rectangles S and give it a subscript three as it is a sum of the area of three rectangles.

S₃ = f(x₁)( x₁ – x₀) + f(x₂)(x₂ – x₁) + f(x₃)(x₃ – x₂)

This can be written more compactly by using summation notation with variable J serving as the subscript and taking values from 1 to 3. The argument to the sum is the height of each rectangle the value of the function x_j multiplied by the width of each rectangle x_j for the x-coordinate of the right side of the j₃ angle minus the x-coordinate of the left side of the same rectangle which is x_j-1. The width can also be written as x_j.

S₃ = )]

Three rectangles do not provide a very good estimate of the area under the curve. We can increase the precision by adding n rectangles i.e. by lowering the width. In this way, the equation may be written as

S_n =  (Reimann Sum).

The upper written equation is called Riemann sum or integral.

S_n only becomes a good estimate of the area under the curve if we take the limit of this quantity as the number of rectangles, the value of n goes to infinity. In the limit of including very many rectangles, S_n is the area between the curve and the x-axis from x₀ to x_n. It is written as symbolically as the integral from x₀ to x_n of the function f(x) multiplied by infinitesimal width dx. This is the Reimann definition of an integral between to fix limits.  It is also called as Reimann integral.

=   (Reimann Integral).

In the integrand, f(X) is the height of an infinitesimally narrow rectangle dx is the infinitesimal width of that rectangle.

Definite integrals:

The difference between the values of integral of a given function f(x) for an upper-value b and lower value a of the independent variable x. These integrals are used to the area over, under, and between the curves. If the area is strictly positive, the area between it and the x-axis is simply the definite integral. If it is negative the area is -1 times the definite integral. We can determine the value of definite integrals by integral calculator.

Indefinite integrals:

Indefinite integrals are the functions whose derivative is equal to the original function f. it is a family of functions whose derivatives are f. Irrespective of definite integrals, the indefinite integral is a function.

 

How can we find integrals?

We can find integrals in two ways.

• Using the manual method.
• Using an anti-derivative or integral calculator.

Manual Method:

Most people use anti derivative calculators however, there is also a method of solving integrals by hand. Let us suppose that we are given the following function:

Where 4 is the upper limit and 3 is the lower limit.

=  +  – 2x

= {  +  – 2(4)} – {  +  – 2(3)}

= 21.333- 7.5

= 13.833

The method discussed in the first example is manual. Now we are going to discuss the other method.

Integral calculators:

Antiderivatives calculator is online software that helps you to find the value of a specific integrand for free. It helps students to solve the most complex integration procedures in less than a minute.

Applications:

• Integrals are of greater use in engineering and physical fields.
• By knowing the function of density we can determine the mass of an object.
• It is of greater importance in Archimida’s principle.