Applications of Integration

Applications of Integration

In my previous posts, we discussed Definite and Indefinite Integrations. Now we shall learn about Applications of Derivatives. Initially, we shall discuss “Area Under Curves”.

Area Under Curve-: If we want to calculate the area between the curves y=f(x) and y=g(x) then there are actually two cases-

First Case when   f(x) \ge g(x)Below is the figure showing this case

Area_G1

here area under these  two curves       

 

The second Case When  f(x) \le g(x)Below figure shows this case

Area_G2

here area between the two curves is      

Both these formulas are the standard formula, we can calculate area using these formulas. But we can see that we are subtracting left side function from Right side function or the below function from the top function. we do so because we want to make sure that we subtract smaller from larger

We can summarize the whole topics in a few steps-

1.The area bounded by the curve y = f(x) , the x-axis and the ordinates at x = a & x = b is given by,             A = \int\limits_a^b {f(x)dx = } \int\limits_a^b {ydx}

2. If the area is below the x-axis then A is negative. The convention is to consider the magnitude only i.e.If the area is below the x-axis then A is negative. The convention is to consider the magnitude only i.e.      A = \left| {\int\limits_a^b {ydx} } \right|

3. Area between the curves y = f (x) & y = g (x) between the ordinates at x = a & x = b is given by,       A = \int\limits_a^b {[f(x) - g(x)]dx}

4. Average value of a function y = f (x) w.r.t. x over an interval    a \le x \le b  is defined as :

                                     {y_{av}} = \frac{1}{{b - a}}\int\limits_a^b {ydx}

Some key Points For JEE Regarding Area Bounded by Curves(applications of integration)


1.Area bounded by an ellipse  \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1   is \pi ab

 

2.The area bounded by an arc of Sinkx and x-axis is or Coskx and x-axis is  \frac{2}{k}

 

3. Area bounded by parabolas {y^2} = 4ax  ,{x^2} = 4ay and x-axis    A = \frac{{5{a^2}}}{4}

 

4. Area bounded by parabolas  {y^2} = 4ax{x^2} = 4by is    A = \frac{{16ab}}{3}

5.Area bounded by parabola  {y^2} = 4ax or  {x^2} = 4ay  and it’s latus rectum x=a is  A = \frac{{8{a^2}}}{3}

 

6.Area bounded by parabola  {y^2} = 4ax (or  {x^2} = 4ay)  and a line y=mx (x=my) is A = \frac{{8{a^2}}}{{3{m^3}}}

Volume of Solids-(applications of integration)

To understand solid of revolution, we first assume a function y=f(x) on an interval [a,b].

VolumeRing_G1

If we rotate this curve around x-axis or y-axis, we get a 3-d solid.If we rotate around x-axis, we get a solid like the one shown below

VolumeRing_G2

Now we shall simply use the formula to find the volume

V = \int\limits_a^b {{A_x}dx}     or if it’s around the y-axis, then the volume     V = \int\limits_a^b {{A_y}dy}

here  {A_x} and {A_y} are the areas of the cross-section of the solid.
Area of cross-section = \pi {r^2}

If it’s in the shape of a ring then area of cross-section= \pi ({R^2} - {r^2})

R and r will depend upon the axis of rotation (either x or y-axis) as well as the given function.

In case the cross section is cylindrical  A = 2\pi rh

Calculation of Work Using Integration(applications of integration) Work done by some force is usually calculated by formula W=F.d here we assume that the force is constant. But If we look at our day to day examples no force is constant.Force is different at every point. If a force any point ‘x’ acts on an object and moves it from x=a to x=b then work done
V = \int\limits_a^b {F.dx}

Calculation of Electrostatic Force Using Integration(applications of integration)-

If we place two charges {q_1} and  {q_2} at some distance ‘x’ then the force between these two charges

f = k\frac{{{q_1}{q_2}}}{{{x^2}}}   we assume that this force is constant but this is never constant.It depends upon the distance between the two charges.

If the distance between the two charges changed from a to b then work done

W = \int\limits_a^b {k\frac{{{q_1}{q_2}}}{{{x^2}}}dx}

Curve Tracing Using Integration(applications of integration): The following outline procedure is to be applied in Sketching the graph of a function y = f (x) which in turn will be extremely useful to quickly and correctly evaluate the area under the curves.

(a) Symmetry: The symmetry of the curve is judged as follows :

(i) If all the powers of y in the equation are even then the curve is symmetrical about the axis of x.

(ii) If all the powers of x are even, the curve is symmetrical about the axis of y.

 

iii) If powers of x & y both are even, the curve is symmetrical about the axis of x as well as y

(iv) If the equation of the curve remains unchanged on interchanging x and y, then the curve is symmetrical about y = x.

(v) If on interchanging the signs of x & y both the equation of the curve is unaltered then there is symmetry in opposite quadrants.
(b) Find dy/dx & equate it to zero to find the points on the curve where you have horizontal tangents.
(c) Find the points where the curve crosses the x-axis & also the y-axis.

(d) Examine if possible the intervals when f (x) is increasing or decreasing. Examine what happens to ‘y’ when  x \to \infty or - \infty

Besides these uses, we also use integration in Chemistry(half-life problems etc.) in History(predicting half-life) and in Economics. Infect integration is base of Economics.

Here is a pdf of practice questions. You can download it here

AREAS OF BOUNDED REGIONS.pdf

 

Calculus is one of the most important areas to crack the JEE Mathematics paper. Here I am sharing all the links to my posts on calculus. You can learn from these posts and download the valuable material using these links:-

Definite Integration

Indefinite Integration

Increasing and Decreasing Functions

Maxima and Minima

Applications of Derivatives in IB Maths(tangents& normals)

Continuity of functions

How To Solve Limit Problems

Limit, Continuity & Differentiability

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