# Karnataka Class 12 Commerce Maths Applications Of The Integrals For Complete Details

## Karnataka Class 12 Commerce Maths Applications Of The Integrals For Complete Details

Karnataka Class 12 Commerce Maths Applications Of The Integrals : Here our team members provides you Karnataka Class 12 Commerce Maths Applications Of The Integrals For Complete Notes in pdf format. Here we gave direct links for you easy to download Karnataka Class 12 Commerce Maths Applications Of The Integrals Notes. Karnataka Class 12 Commerce Maths Applications Of The Integrals topics are Applications in finding the area under simple curves, especially lines, circles/parabolas/ellipses (in standard form only), Area between any of the two above said curves (the region should be clearly identifiable). Download these Karnataka Class 12 Commerce Maths Applications Of The Integrals For Complete Notes and read well.

### Karnataka Class 12 Commerce Maths Applications Of The Integrals For Complete Details

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Karnataka Class 12 Commerce Maths Applications Of The Integrals : In geometry, we have learnt formulae to calculate areas of various geometrical figures including triangles, rectangles, trapezias and circles. Such formulae are fundamental in the applications of mathematics to many real life problems. The formulae of elementary geometry allow us to calculate areas of many simple figures. However, they are inadequate for calculating the areas enclosed by curves. For that we shall need some concepts of Integral Calculus. In the previous chapter, we have studied to find the area bounded by the curve y = f (x), the ordinates x = a, x = b and x-axis, while calculating definite integral as the limit of a sum. Here, in this chapter, we shall study a specific application of integrals to find the area under simple curves, area between lines and arcs of circles, parabolas and ellipses (standard forms only). We shall also deal with finding the area bounded by the above said curves.

### Karnataka Class 12 Commerce Maths Applications Of The Integrals For Complete Details

Karnataka Class 12 Commerce Maths Applications Of The Integrals : In the previous chapter, we have studied definite integral as the limit of a sum and how to evaluate definite integral using Fundamental Theorem of Calculus. Now, we consider the easy and intuitive way of finding the area bounded by the curve y = f (x), x-axis and the ordinates x = a and x = b. From Fig 8.1, we can think of area under the curve as composed of large number of very thin vertical strips. Consider an arbitrary strip of height y and width dx, then dA (area of the elementary strip)= ydx, where, y = f(x).

This area is called the elementary area which is located at an arbitrary position within the region which is specified by some value of x between a and b. We can think of the total area A of the region between x-axis, ordinates x = a, x = b and the curve y = f (x) as the result of adding up the elementary areas of thin strips across the region PQRSP.

#### Chapter Contents

1. Applications of the Indefinite Integral shows how to find displacement (from velocity) and velocity (from acceleration) using the indefinite integral. There are also some electronics applications in this section.

In primary school, we learned how to find areas of shapes with straight sides (e.g. area of a triangle or rectangle). But how do you find areas when the sides are curved? We’ll find out how in:

• 2. Area Under a Curve and
• 3. Area Between 2 Curves

4. Volume of Solid of Revolution explains how to use integration to find the volume of an object with curved sides, e.g. wine barrels.

5. Centroid of an Area means the centre of mass. We see how to use integration to find the centroid of an area with curved sides.

6. Moments of Inertia explains how to find the resistance of a rotating body. We use integration when the shape has curved sides.

7. Work by a Variable Force shows how to find the work done on an object when the force is not constant. This section includes Hooke’s Law for springs.

### Survival Tips

Before you start this section, it’s a good idea to revise:

• Graph of the Quadratic Function
• Graphs of Exponential and Log Functions
• Plane Analytic Geometry
• Curve Sketching

(This chapter is easier if you can draw curves confidently.)

You may also wish to see the Introduction to Calculus.

8. Electric Charges have a force between them that varies depending on the amount of charge and the distance between the charges. We use integration to calculate the work done when charges are separated.

9. Average Value of a curve can be calculated using integration.

Head Injury Criterion is an application of average value and used in road safety research.

10. Force by Liquid Pressure varies depending on the shape of the object and its depth. We use integration to find the force.

In each case, we solve the problem by considering the simple case first. Usually this means the area or volume has straight sides. Then we extend the straight-sided case to consider curved sides. We need to use integration because we have curved sides and cannot use the simple formulas any more.

### Karnataka Class 12 Commerce Maths Applications Of The Integrals For Complete Details

Karnataka Class 12 Commerce Maths Applications Of The Integrals : In this last chapter of this course we will be taking a look at a couple of applications of integrals.  There are many other applications, however many of them require integration techniques that are typically taught in Calculus II.  We will therefore be focusing on applications that can be done only with knowledge taught in this course.

Because this chapter is focused on the applications of integrals it is assumed in all the examples that you are capable of doing the integrals.  There will not be as much detail in the integration process in the examples in this chapter as there was in the examples in the previous chapter.

#### Here is a listing of applications covered in this chapter.

Average Function Value  We can use integrals to determine the average value of a function.

Area Between Two Curves  In this section we’ll take a look at determining the area between two curves.

Volumes of Solids of Revolution / Method of Rings  This is the first of two sections devoted to find the volume of a solid of revolution.  In this section we look at the method of rings/disks.

Volumes of Solids of Revolution / Method of Cylinders  This is the second section devoted to finding the volume of a solid of revolution.  Here we will look at the method of cylinders.

More Volume Problems  In this section we’ll take a look at finding the volume of some solids that are either not solids of revolutions or are not easy to do as a solid of revolution.

Work  The final application we will look at is determining the amount of work required to move an object.

### Karnataka Class 12 Commerce Maths Applications Of The Integrals For Complete Details

Karnataka Class 12 Commerce Maths Applications Of The Integrals : In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two main operations of calculus, with its inverse, differentiation, being the other. Given a function f of a real variable x and an interval [a, b] of the real line, the definite integral

1) Why do we need integration?

In physics, integration crops up pretty much everywhere. Work is the integral of force over a distance, for example. Electric flux is an integral of the electric field over a surface. In other sciences, you might want to compute the area under a curve. (Don’t re-invent calculus like this though). In pure math, integrals are used for concepts such as winding numbers and are irreplaceable for results such as the general Stokes’ theorem.

2) Do we have integrals in multi variable calculus? Is there any practical use of integration?

Absolutely. See multiple integral, line integral, surface integral, contour integral (admittedly, a particular type of line integral, but it holds special importance).

3) What is the most important prerequisite for Stochastic calculus?

Calculus and probability theory (not statistics!)

#### There are two methods for finding the area bounded by curves in rectangular coordinates. These are…

1. by using a horizontal element (called strip) of area, and
2. by using a vertical strip of area.

The strip is in the form of a rectangle with area equal to length × width, with width equal to the differential element. To find the total area enclosed by specified curves, it is necessary to sum up a series of rectangles defined by the strip.

#### Using Horizontal Strip

From the figure, the area of the strip is xdyxdy, where x=xRxLx=xR−xL. The total area can be found by running this strip starting from y1y1 going to y2y2. Our formula for integration is…
A=y2y1xdy=y2y1(xRxL)dyA=∫y1y2xdy=∫y1y2(xR−xL)dy

Note that xRxR is the right end of the strip and is always on the curve f(y)f(y) and xLxL is the left end of the strip and is always on the curve g(y)g(y). We therefore substitute xR=f(y)xR=f(y)and xL=g(y)xL=g(y) prior to integration.

#### Using Vertical Strip

We apply the same principle of using horizontal strip to the vertical strip. Consider the figure below.

The total area is…
A=x2x1ydx=x1x2(yUyL)dxA=∫x1x2ydx=∫x1x2(yU−yL)dx

Where
yUyU = upper end of the strip = f(x)f(x)
yLyL = lower end of the strip = g(x)g(x)

The steps in finding the area can be outlined as follows:
1. Sketch the curve
2. Decide what strip to use and define its limits
3. Apply the appropriate formula based on the strip then integrate.

### Karnataka Class 12 Commerce Maths Applications Of The Integrals For Complete Details

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