xڕS�R�0��+t�j�:fXRp��s"�G8�b�����}Z�L!S��AO��~�V��ъX/��������_�{�3x��Qin�A*����)���P���L The integral of the flow rate 2x tells us the volume of water: And the slope of the volume increase x2+C gives us back the flow rate: And hey, we even get a nice explanation of that "C" value ... maybe the tank already has water in it! 0000007435 00000 n
Integration is a way of adding slices to find the whole. /Parent 9 0 R 0000001531 00000 n
/Length 386 differentiation means difference -division or integration means product sum so here division reverse product (multiplication) difference reverse sum so we can write differentiation = dy/dx or integration =. /Type /XObject Integration is covered in tutorial 1. We always differentiate a function with respect to a variable because the change is always relative. Example: with the flow in liters per minute, and the tank starting at 0 . 0000033222 00000 n
Integrals of Trig. �;|)�dX����>���V'Dz�������2v��%�����݅!�KQ�[LD͆g��|�"��2�U(y%�3s��a ��E+�T$k&Fę�#�.`⮞��Uy96����UO2E+�
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On completion of this tutorial you should be able to do the following. we have study some basic concept of calculus in previous post continuing that post ahead we will study about differentiation and integration concept in this post lets start. Each is the reverse process of the other. 2 • We have seen two applications: – signal smoothing – root finding • Today we look – differentation – integration 6 0 obj
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The symbol for "Integral" is a stylish "S" 0000007414 00000 n
now there are two type of integration one is indefinite integral and other is definite integral when we write ⨜ydx = ⨜f(x)dx or w = ⨜(3x-2)dx after solving we will get 3x²/2 -2x +c hence work done w = 3x²/2 -2x +c where c is constant of integration which is again a function of x no exact value of work done how much it depend upon value of x so it is indefinite integral because it may be any value for different value of x means no certain value but for definite integral its value is certain and limit is given between the range lower to upper limit so for definite integral we will get definite value like 10, 20, 100, 200..hence it is called definite integral thanks for reading . 0000040682 00000 n
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As the flow rate increases, the tank fills up faster and faster. Your email address will not be published. 0000006799 00000 n
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But remember to add C. From the Rules of Derivatives table we see the derivative of sin(x) is cos(x) so: But a lot of this "reversing" has already been done (see Rules of Integration). s = 3t4 • Reduce the old power by one and use this as the new power. (for "Sum", the idea of summing slices): After the Integral Symbol we put the function we want to find the integral of (called the Integrand). Explain differential coefficients. Integration: With a flow rate of 2x, the tank volume increases by x 2. Now integration is reverse process of differentiation how ? So when we reverse the operation (to find the integral) we only know 2x, but there could have been a constant of any value. ]�gLau��U.��� �E����#���^'ܤrc����x���-��ܧ�/ob��'�����&�������]���iW�!Ռy�(�SŏO�|(��8��z:���]���KZK/���L��y�\�Y��/k��@�c\��o�x� �y��}�}j!� |��t /Filter /FlateDecode 0000002579 00000 n
A Definite Integral has actual values to calculate between (they are put at the bottom and top of the "S"): At 1 minute the volume is increasing at 2 liters/minute (the slope of the volume is 2), At 2 minutes the volume is increasing at 4 liters/minute (the slope of the volume is 4), At 3 minutes the volume is increasing at 6 liters/minute (a slope of 6), The flow still increases the volume by the same amount. Differentiation and Integration Basics Integration differentiation are two different parts of calculus which deals with the changes. Apply Newton’s rules of differentiation to basic functions. But it is easiest to start with finding the area under the curve of a function like this: We could calculate the function at a few points and add up slices of width Δx like this (but the answer won't be very accurate): We can make Δx a lot smaller and add up many small slices (answer is getting better): And as the slices approach zero in width, the answer approaches the true answer. Integration is covered in tutorial 1. /Type /Page 0000064074 00000 n
Volume and Capacity. Integration can be used to find areas, volumes, central points and many useful things. And the increase in volume can give us back the flow rate. now there are two type of integration one is indefinite integral and other is definite integral when we write, ⨜(3x-2)dx after solving we will get 3x²/2 -2x +c hence work done w =. now from equation put t = 0 then x = 1 again put t = 5 then x = 51, Average velocity = (51-1)/(5-0) = 50/5 = 10 m/s this is average velocity but we have to find the instantaneous velocity at t = 5 s hence for this we have to use differentiation of x with respect to t, we know general formula for differentiation dy/dx = x^n = nx^(n-1). Well, we have played with y=2x enough now, so how do we integrate other functions? And learn something new in this article which you don’t know about Quantum Physics. Derivative: If the tank volume increases by x 2, then the flow rate must be 2x. change of function y division with respect to x is called differentiation physical meaning is slope y may be function of any variable time, displacement why differentiation is important and how we use differentiation for this we will take an example to better understand s is displacement of particle dependent upon time equation given x = 2t² +1 now we want to find velocity and displacement after 5 second. >>/Font << /F1 12 0 R/F2 13 0 R>> 0000004007 00000 n
H��SMK�@{��0�xp���٫ъ�Bi��x�ږL� ��;���ji+�Bfg2;�ޛ�4�. /BBox [0.00000000 0.00000000 529.22500000 166.15000000] ⨜x^ndx = x^(n+1)/(n+1) remember this basic formula. /GS1 14 0 R 0000001756 00000 n
/ExtGState << Imagine the flow starts at 0 and gradually increases (maybe a motor is slowly opening the tap). 0000005072 00000 n
⨜ydx hence these two are reverse process of each other in physics we use both wherever application required . NCERT Solutions. So that you can never forget. This book is a revised and expanded version of the lecture notes for Basic Calculus and other similar courses o ered by the Department of Mathematics, University of Hong Kong, from the first semester of the academic year 1998-1999 through the second semester of 2006-2007. 0000005705 00000 n
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whenever we differentiate a quantity then other quantity is made like displacement differentiation with respect to time give velocity, velocity differentiation with respect to time gives acceleration. /Subtype /Form /PTEX.PageNumber 1 Each is the reverse process of the other. Integrating the flow (adding up all the little bits of water) gives us the volume of water in the tank. For a given function, y = f(x), continuous and defined in
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