{"id":524,"date":"2017-10-22T00:42:07","date_gmt":"2017-10-21T19:12:07","guid":{"rendered":"http:\/\/ibelitetutor.com\/blog\/?p=524"},"modified":"2023-08-14T12:45:27","modified_gmt":"2023-08-14T07:15:27","slug":"applications-of-integrations","status":"publish","type":"post","link":"https:\/\/ibelitetutor.com\/blog\/applications-of-integrations\/","title":{"rendered":"Applications of Integration"},"content":{"rendered":"

Applications of Integration<\/strong><\/h2>\n

In my previous posts, we discussed Definite and Indefinite Integrations. Now IB Maths Tutors<\/a><\/strong> will learn about Applications of Derivatives. Initially, we shall discuss “Area Under Curves”.<\/p>\n

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

First Case when \u00a0\u00a0\"f(x)– <\/strong>Below is the figure showing this case<\/p>\n

\"Area_G1\"<\/p>\n

here area under these \u00a0two curves \u00a0 \u00a0 \u00a0\u00a0<\/p>\n

The second Case When \u00a0\"f(x)– <\/strong>Below figure shows this case<\/p>\n

\"Area_G2\"<\/p>\n

<\/p>\n

here area between the two curves is \u00a0 \u00a0 \u00a0<\/p>\n

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<\/strong> from Right side function<\/strong> or the below function<\/strong> from the top function<\/strong>. we do so because we want to make sure that we subtract smaller from larger<\/strong><\/p>\n

We can summarize the whole topics in a few steps-<\/p>\n

1.The area bounded by the curve y = f(x) , the x-axis and the ordinates at x = a & x = b is given by, \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\"A<\/p>\n

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. \u00a0 \u00a0 \u00a0\"A<\/p>\n

3. Area between the curves y = f (x) & y = g (x) between the ordinates at x = a & x = b is given by, \u00a0 \u00a0 \u00a0\u00a0\"A<\/p>\n

4.\u00a0Average value of a function y = f (x) w.r.t. x over an interval \u00a0 \u00a0\"a\u00a0\u00a0<\/span>is defined as :<\/span><\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\"{y_{av}}<\/p>\n

Some key Points For JEE Regarding Area Bounded by Curves(applications of integration)<\/strong><\/p>\n


\n<\/strong>1.Area bounded by an ellipse \u00a0\"\\frac{{{x^2}}}{{{a^2}}}\u00a0 \u00a0is\u00a0\"\\pi<\/p>\n

2.The area bounded\u00a0by an arc of Sinkx\u00a0and x-axis is or Coskx and x-axis is \u00a0\"\\frac{2}{k}\"<\/p>\n

3. Area bounded by parabolas\u00a0\"{y^2}\u00a0 ,\"{x^2}\u00a0and x-axis \u00a0 \u00a0\"A<\/p>\n

4. Area bounded by parabolas \u00a0\"{y^2},\u00a0\"{x^2}\u00a0is \u00a0 \u00a0\"A<\/p>\n

5.Area bounded by parabola \u00a0\"{y^2}\u00a0or \u00a0\"{x^2}\u00a0\u00a0and it’s latus rectum x=a is \u00a0\"A<\/p>\n

6.Area bounded by parabola \u00a0\"{y^2}\u00a0(or \u00a0\"{x^2})\u00a0\u00a0and a line y=mx (x=my) is \"A<\/p>\n

Volume of Solids-(applications of integration)<\/strong><\/p>\n

To understand solid of revolution, we first assume a function y=f(x) on an interval [a,b<\/em>].<\/p>\n

\"VolumeRing_G1\"<\/p>\n

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<\/p>\n

Now we shall simply use the formula to find the volume<\/p>\n

\"V\u00a0 \u00a0 or if it’s around the y-axis, then the volume \u00a0 \u00a0\u00a0\"V<\/p>\n

<\/p>\n

here \u00a0\"{A_x}\"\u00a0and\u00a0<\/span>\"{A_y}\"\u00a0are the areas of the cross-section of the solid.
\nArea of cross-section =\u00a0\"\\pi<\/p>\n

If it’s in the shape of a ring then area of cross-section=\u00a0\"\\pi<\/p>\n

R and r will depend upon the axis of rotation (either x or y-axis) as well as the given function.<\/p>\n

In case the cross section is cylindrical \u00a0\"A<\/p>\n

Calculation of Work Using Integration(applications of integration)<\/strong> Work done by some force is usually calculated by formula W=F.d<\/strong> 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
\n\"V<\/p>\n

Calculation of Electrostatic Force Using Integration(applications of integration)- <\/strong><\/p>\n

If we place two charges\u00a0\"{q_1}\"\u00a0and \u00a0<\/span>\"{q_2}\"\u00a0at some distance ‘x’ then the force between these two charges<\/span><\/p>\n

\"f\u00a0 \u00a0we assume that this force is constant but this is never constant. It depends upon the distance between the two charges.<\/p>\n

If the distance between the two charges changed from a<\/strong> to b<\/strong> then work done<\/p>\n

\"W<\/p>\n

Curve Tracing Using Integration(applications of integration):<\/strong> 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.<\/p>\n

(a)<\/span> Symmetry: The symmetry of the curve is judged as follows :<\/p>\n

(i) If all the powers of y in the equation are even then the curve is symmetrical about the axis of x.<\/p>\n

(ii) If all the powers of x are even, the curve is symmetrical about the axis of y.<\/p>\n

iii) If powers of x & y both are even, the curve is symmetrical about the axis of x as well as y<\/p>\n

(iv) If the equation of the curve remains unchanged on interchanging x and y, then the curve is symmetrical about y = x.<\/p>\n

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

(d) Examine if possible the intervals when f (x) is increasing or decreasing. Examine what happens to \u2018y\u2019 when \u00a0\"x<\/p>\n

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.<\/p>\n

<\/h4>\n

\u200b<\/h4>\n

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:-<\/p>\n

<\/p>\n

Definite Integration<\/h4>\n

Indefinite Integration<\/h4>\n

Increasing and Decreasing Functions<\/h4>\n

Maxima and Minima<\/h4>\n

Applications of Derivatives in IB Maths(tangents& normals)<\/h4>\n

Continuity of functions<\/h4>\n

How To Solve Limit Problems<\/h4>\n

Limit, Continuity & Differentiability<\/h4>\n

\"ib<\/p>\n\n

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    <\/ul><\/div>\n
    \n
    \n<\/fieldset>\n

    \n<\/p>

    <\/div>\n<\/form>\n<\/div>\n\n

    <\/p>\n

    <\/p>\n

    <\/p>\n

    <\/p>\n","protected":false},"excerpt":{"rendered":"

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