APPLICATIONS OF INTEGRALS

Today we will talk about the applications of integrals. And we will see some examples.

Let's go!

Both the integral calculus and differential calculus are part of what is known as infinitesimal calculus, this being an important constituent of modern mathematics. 

Focusing on the integral calculus, this is the one that consist of the reverse process to derivate having a main applications or uses, which are the calculation of areas, volumes and length of curve.

Beginning with the calculation of flat areas, you must first mention that the result obtained from the integral can be both positive and negative, null, while the value of an area must always be positive, an area can not be negative , which causes is that when calculating the sign area zones whose areas we want to determine, since it must take the same value at all to add them later to be taken into account.

Although this indicated, in practice it is always more comfortable and efficient draw graphs for functions, calculate the intersection points between them and add their integrals to obtain the result.

Calculating volumes. When the object rotates about an axis, it is called: a solid of revolution, and the axis, the axis of revolution. There are different methods or formulas to calculate volume using integrals.

Some of these methods are: the disk method, which is used when what rotates around the axis is a rectangle, considering that forming a cylinder, and has certain formulas to be resolved:

(where r is a function of x), the method of the washers, the solids used when revolution have a hole, whose formula solved by the limited integrated between the two points of the number π for f (x) squared minus g (x) squared, and finally the method of known sections, which is used when you know the area of bases partial cylinders that have divided the solid.

These are not the only possible methods of performing a calculation using volume integrals.

As an example of using some exercises integral to calculate areas and volumes:

1. Find the area bounded by the line x + y = 10, the x-axis and the ordinates x = 2 and x = 8.

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b) Calculate the area bounded by the graphs of the functions y^2 = 4x and y = x^2.

c) Calculate the area bounded by the parabola y = x^2 + 2 and the line passing through the points (-1, 0) and (1, 4).

d) Find the area of the space bounded by the parabola y = 4x - x2 and the tangent to the curve at the points of intersection with the x axis.

· Intersection points:

· Equation of the tangent to the parabola at the point (0, 0):

· Equation of the tangent to the parabola at the point (4, 0):

2. Find the volume of the trunk cone generated by rotation about OX the area bounded by 

y = 6 - x, y = 0, x = 0, x = 4.

b) Calculate the volume that generates a triangle with vertices A (3, 0), B (6, 3), C (8, 0) to rotate 

360 ​​° around the axis OX.

· Equation of the line through AB:

· Equation of the line through BC:

c) Find the volume of revolution generated by rotating the body about the axis OX, the region determined by the function f (x) = 1/2 + cos x, the horizontal axis and the lines x = 0 and x = π.

d) Find the volume generated by the circle x2 + y2 - 4x = -3 to rotate about the axis OX.

3. The center of the circle is C (0, 1) and the radius r = 1.

Breakpoints with the OX axis:

The following application of the integral is the length of curve. This is to approximate an arc, a piece of curved, by straight segments. Calculating the length of curve is given by a set formula that is:

This formula is obtained when f defined on [a, b] is derivable, and if f 'in same range is continuous.

4. Example curve length:

a) Find the length of the arc of a curvein the interval [0, 1].

These integral calculus and its applications can be applied to both the physical and engineering especially for calculations such as calculating the area of ​​a piece.

Hope this has been helpful!