tag:blogger.com,1999:blog-136031501056944216.post3708552912114240888..comments2023-09-15T12:44:26.510+05:00Comments on Virtual Study Solutions: MTH101 short notes or past paperMr Khanhttp://www.blogger.com/profile/04580984118785378337noreply@blogger.comBlogger9125tag:blogger.com,1999:blog-136031501056944216.post-32798901402054023062021-02-23T13:55:46.394+05:002021-02-23T13:55:46.394+05:00Lecture No 23 to 45
Complete and Important Questio...Lecture No 23 to 45<br />Complete and Important Question and answer<br />1. What is the difference between definite integral and an indefinite integral?<br />If the function f is continuous on [a, b], and can assume both<br />positive and negative values, the definite integral<br />( )<br />b<br />a<br />f x dx <br />is net signed area between y = f(x) and the interval [a, b]. The<br />numbers a and b are called the lower and upper limits of<br />integration.<br />Indefinite integral is the set of functions F(x) + C, where C is constant of integration and F(x) is the<br />integral of given function f where<br /> ( )<br />( )<br />d F x<br />f x<br />dx<br /> .<br />Q 2. What is meant by net signed area (the term used in defining definite integral)?<br />An indefinite integral is of the form<br />f x dx F x C ( ) ( ) <br />Suppose that a function f has a smooth curve in an interval [a, b] and can lie both above and below xaxis then ‘area under the curve and above x-axis’ minus ‘area above the curve below x-axis’ is termed as<br />net signed area under the curve of f on the interval [a, b]. It is named "signed" area<br />because the area above the x -axis counts as positive and the area below the x -axis<br />counts as negative.Net signed area can be positive, negative or zero; it is positive<br />when there is more area above than below, negative when there is more area<br />below than above, and zero when the area above and below are equal.For example,<br />in the figure<br />Net signed area under the curve of f on the interval [a, b]<br />Visit for more... www.vustudy.com<br /><br />= [Area above] – [Area below]<br />= (A + C) – B<br />Q 3. Why we add C with the anti-derivative for evaluating an indefinite integral?<br />ANSWER: If a function f(x) is defined on an interval and F(x) is an antiderivative of f(x),<br />then the indefinite integral is set of all anti-derivatives of f(x), that is, the functions F(x) + C,<br />where C is an arbitrary constant.<br />As we know, the derivative of any constant function is zero. Once one has found one<br />antiderivative F(x), adding or subtracting a constant C will give us another antiderivative,<br />because (F(x) + C) ' = F ' (x) + C ' = F ' (x) . The constant is a way of expressing that<br />every function has an infinite number of different anti-derivatives.<br />For example, suppose one wants to find anti-derivatives of cos(x). One such anti-derivative is<br />sin(x). Another one is sin(x) + 1. A third is sin(x) − π. Each of these has derivative<br />cos(x), so they are all anti-derivatives of cos(x).<br />It turns out that adding and subtracting constants is the only flexibility we have in finding<br />different anti-derivatives of the same function. That is, all anti-derivatives are the same up to a<br />constant. To express this fact for cos(x), we write:<br />∫ cos (x) dx = sin(x) + C<br />Q 6. State First Fundamental Theorem of Calculus?<br />The First Fundamental Theorem of Calculus states that:<br />If f is continuous on the closed interval [a, b] and F is the anti-derivative of f on<br />[ , ] a b<br />, then<br />( ) ( ) ( )<br />b<br />a<br />f x dx F b F a <br />Q.1# Why we use the second fundamental theorem of calculus?<br /><br />Answer: The first fundamental theorem of calculus is used to evaluate the definite integral of a<br />continuous function if we can find an antiderivative for that function, but it does not point out<br />the question of which functions actually have antiderivatives; then at this stage the second<br />fundamental theorem of calculus is used.<br />The second fundamental theorem of calculus states that the derivative of a definite integral<br />with respect to its upper endpoint is its integrand; it allows one to compute the definite integral<br />of a function by using any one of its infinitely many antiderivatives. This part of the theorem has<br />invaluable practical applications, because it markedly simplifies the computation of definite<br />integral.<br /><br /><br /><br />Anonymoushttps://www.blogger.com/profile/01643824844985806147noreply@blogger.comtag:blogger.com,1999:blog-136031501056944216.post-70048146102025502252021-02-23T13:53:27.746+05:002021-02-23T13:53:27.746+05:00Complete and Important Question and answer
1. What...Complete and Important Question and answer<br />1. What is the difference between definite integral and an indefinite integral?<br />If the function f is continuous on [a, b], and can assume both<br />positive and negative values, the definite integral<br />( )<br />b<br />a<br />f x dx <br />is net signed area between y = f(x) and the interval [a, b]. The<br />numbers a and b are called the lower and upper limits of<br />integration.<br />Indefinite integral is the set of functions F(x) + C, where C is constant of integration and F(x) is the<br />integral of given function f where<br /> ( )<br />( )<br />d F x<br />f x<br />dx<br /> .<br />Q 2. What is meant by net signed area (the term used in defining definite integral)?<br />An indefinite integral is of the form<br />f x dx F x C ( ) ( ) <br />Suppose that a function f has a smooth curve in an interval [a, b] and can lie both above and below xaxis then ‘area under the curve and above x-axis’ minus ‘area above the curve below x-axis’ is termed as<br />net signed area under the curve of f on the interval [a, b]. It is named "signed" area<br />because the area above the x -axis counts as positive and the area below the x -axis<br />counts as negative.Net signed area can be positive, negative or zero; it is positive<br />when there is more area above than below, negative when there is more area<br />below than above, and zero when the area above and below are equal.For example,<br />in the figure<br />Net signed area under the curve of f on the interval [a, b]<br />Visit for more... www.vustudy.com<br /><br />= [Area above] – [Area below]<br />= (A + C) – B<br />Q 3. Why we add C with the anti-derivative for evaluating an indefinite integral?<br />ANSWER: If a function f(x) is defined on an interval and F(x) is an antiderivative of f(x),<br />then the indefinite integral is set of all anti-derivatives of f(x), that is, the functions F(x) + C,<br />where C is an arbitrary constant.<br />As we know, the derivative of any constant function is zero. Once one has found one<br />antiderivative F(x), adding or subtracting a constant C will give us another antiderivative,<br />because (F(x) + C) ' = F ' (x) + C ' = F ' (x) . The constant is a way of expressing that<br />every function has an infinite number of different anti-derivatives.<br />For example, suppose one wants to find anti-derivatives of cos(x). One such anti-derivative is<br />sin(x). Another one is sin(x) + 1. A third is sin(x) − π. Each of these has derivative<br />cos(x), so they are all anti-derivatives of cos(x).<br />It turns out that adding and subtracting constants is the only flexibility we have in finding<br />different anti-derivatives of the same function. That is, all anti-derivatives are the same up to a<br />constant. To express this fact for cos(x), we write:<br />∫ cos (x) dx = sin(x) + C<br /><br />Anonymoushttps://www.blogger.com/profile/01643824844985806147noreply@blogger.comtag:blogger.com,1999:blog-136031501056944216.post-45558562687240598292020-07-01T20:14:54.740+05:002020-07-01T20:14:54.740+05:00Mth 101 vu MCQs are required for preparation mid ...Mth 101 vu MCQs are required for preparation mid /grand quiz spring 2020.plz mail me Anonymoushttps://www.blogger.com/profile/00853955632784733566noreply@blogger.comtag:blogger.com,1999:blog-136031501056944216.post-882767801572281502019-12-24T14:48:58.559+05:002019-12-24T14:48:58.559+05:00Math101 ke important question hain Apke pass?Math101 ke important question hain Apke pass?آسان بات https://www.blogger.com/profile/11528138109069243211noreply@blogger.comtag:blogger.com,1999:blog-136031501056944216.post-55754527207037976272019-12-24T14:48:08.962+05:002019-12-24T14:48:08.962+05:00Aoa..Aoa..آسان بات https://www.blogger.com/profile/11528138109069243211noreply@blogger.comtag:blogger.com,1999:blog-136031501056944216.post-48672885359438840142019-12-19T07:17:30.521+05:002019-12-19T07:17:30.521+05:00mid term ky notes chahiye ye to lecture 32 sy star...mid term ky notes chahiye ye to lecture 32 sy start ho rhyAnonymoushttps://www.blogger.com/profile/08016943673922253154noreply@blogger.comtag:blogger.com,1999:blog-136031501056944216.post-29297348107552240212019-12-11T16:43:58.191+05:002019-12-11T16:43:58.191+05:00mth101 k short notes chahiye plZzzz koi snd kr dy
...mth101 k short notes chahiye plZzzz koi snd kr dy<br />Anonymoushttps://www.blogger.com/profile/07797681416000297103noreply@blogger.comtag:blogger.com,1999:blog-136031501056944216.post-24471174568056290062019-12-05T17:07:06.644+05:002019-12-05T17:07:06.644+05:00very helpfulvery helpfulAnonymoushttps://www.blogger.com/profile/17250371387919588362noreply@blogger.comtag:blogger.com,1999:blog-136031501056944216.post-45008927330039144362013-05-04T15:46:39.197+05:002013-05-04T15:46:39.197+05:00Hania gud keep it up & thanks for sharingHania gud keep it up & thanks for sharing <br />Esha maliknoreply@blogger.com