{"id":507,"date":"2017-10-08T02:07:50","date_gmt":"2017-10-07T20:37:50","guid":{"rendered":"http:\/\/ibelitetutor.com\/blog\/?p=507"},"modified":"2023-08-14T12:54:25","modified_gmt":"2023-08-14T07:24:25","slug":"definite-integration-ib-mathematics","status":"publish","type":"post","link":"http:\/\/ibelitetutor.com\/blog\/definite-integration-ib-mathematics\/","title":{"rendered":"Definite Integration-Topics in IB Mathematics"},"content":{"rendered":"

Definite Integration<\/strong><\/h2>\n

In the previous post, our IB Maths Tutors<\/span> discussed indefinite integration.<\/span> Now we shall discuss definite integration<\/p>\n

\u25ba Definite Integration-\u00a0<\/strong><\/span>We already know that \u00a0\u00a0\"\\int\u00a0\u00a0<\/span><\/span><\/span>\"<\/span>\u00a0this c here is an integral constant. we are not sure about its value. This c is the reason we call this process\u00a0indefinite integration. But suppose we do our integration between certain limits like:-<\/p>\n

\"\\int\\limits_a^b\u00a0 \u00a0here a\"\u00a0lower limit while b\"\u00a0higher limit<\/p>\n

<\/p>\n

\"\\int\\limits_a^b <\/p>\n

=g(b)-g(a)<\/p>\n

You can clearly see that this function is independent of ‘c’. Means we can be sure about its value so this type of integration is called \u00a0Definite\u00a0Integration<\/strong>.<\/p>\n

\u25baDefinite\u00a0Integration of a function f(x) is possible in [a,b] if f(x) is continuous in the given interval<\/p>\n

\u25baIf f(x), the integrand, is not continuous for a given value of x then it doesn’t mean that g(x), the integral, is also discontinuous for that value of x.<\/p>\n

\u25ba Definite integration of a function between given limits like \u00a0 \u00a0\u00a0\"\\int\\limits_a^b\u00a0 \u00a0 \u00a0 \u00a0 Algebraic sum of areas bounded by the given curve f(x) and given lines x=a and x=b. That’s why the answer for definite integration problems is a single number.<\/p>\n

\u25ba\u00a0If\u00a0\"\\int\\limits_a^b\u00a0that shows a few things:-<\/p>\n

(i) The lines between which area is bounded are co-incident(a=b)<\/p>\n

(ii) Area covered above the x-axis=Area covered below the x-axis that means positive part of area and negative part of area is equal<\/p>\n

(iii) there must be at least one solution\/root to f(x) between x=a and x=b(this is something we study in ROLE’S THEOREM in detail)<\/p>\n

\u25ba If given function f(x) is not continuous at x=c then we should write<\/p>\n

\"\\int\\limits_a^b<\/p>\n

\u25ba If given function f(x) > or <0 in any given interval (a,b) then \u00a0\"\\int\\limits_a^b\u00a0>0 or <0 in given interval (a,b)<\/p>\n

\u25ba\u00a0If given function f(x)\u00a0\"\u00a0g(x) in the given interval (a,b) then \u00a0 \u00a0\"\\int\\limits_a^b\u00a0<\/span><\/p>\n

in the given interval<\/span><\/p>\n

\u25ba If we integrate the given function f(x) in the given interval (a,b) then<\/p>\n

\"\\int\\limits_a^b<\/p>\n

 <\/p>\n

<\/strong><\/p>\n

Some More Properties of Definite Integration:-<\/strong><\/p>\n

1.\u00a0\u00a0 We can interchange the limits on any definite integral, all that we need to do is tack a minus sign onto the integral when we do.<\/p>\n

2.\u00a0.\u00a0 If the upper and lower limits are the equal then integration of function will be zero
\n3.\u00a0\u00a0, where\u00a0c<\/em>\u00a0is any constant\/any real number<\/p>\n

4.\u00a0\u00a0 that means definite integration is a distributive process<\/p>\n

5. \u00a0\u00a0 here c is a number lying somewhere between a and b<\/p>\n

6. \u00a0\u00a0 If we don’t\u00a0change the integrand and the limits, then change in the variable will not affect the answer<\/p>\n

7.(a) If f(x) is an odd function i.e. f(x) = – f(-x) then \u00a0\u00a0\"\\int\\limits_{<\/p>\n

(b) \u00a0If f(x) is an even function i.e. f(x) = f(-x) then \u00a0 \u00a0\u00a0\"\\int\\limits_{<\/p>\n

8. \u00a0\u00a0\"\\int\\limits_a^b\u00a0 \u00a0in particular \u00a0 \u00a0\"\\int\\limits_0^a<\/p>\n

9. \u00a0\u00a0\"\\int\\limits_0^a<\/p>\n

10. \"\\int\\limits_{ma}^{na}\u00a0where f(a) is periodic with period ‘a’.<\/span><\/p>\n

Walli\u2019s Formula:\u00a0<\/strong><\/p>\n

\"\\int\\limits_0^{\\pi=\u00a0\"\\frac{{\\left[<\/p>\n

Where K =\"\\pi\u00a0 \u00a0if both m and n are even \u00a0 (m, n\"\u00a0N)<\/p>\n

= 1 in case the function is odd<\/p>\n

Here<\/p>\n

Leibnitz’s\u00a0Rule- <\/strong>If f(x) is a continuous function and u(x) & v(x) are differentiable in the interval [a,b] then,<\/p>\n

\"\\frac{d}{{dx}}\\int\\limits_{u(x)}^{v(x)}<\/p>\n

This rule is used when at least one of the limits is a function.<\/p>\n

Here is a very detailed Pdf for definite integration download it solves the questions<\/p>\n

<\/div>\n
Here are the links to articles written on calculus so far, you should read them for better understanding of calculus<\/div>\n
<\/div>\n
\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

\n

<\/p>

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

    \n<\/p>

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

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

    Definite Integration In the previous post, our IB Maths Tutors discussed indefinite integration. Now we shall discuss definite integration \u25ba Definite Integration-\u00a0We already know that […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[11,5],"tags":[],"class_list":["post-507","post","type-post","status-publish","format-standard","hentry","category-cbse-tutors","category-ib-mathematics-tutors"],"_links":{"self":[{"href":"http:\/\/ibelitetutor.com\/blog\/wp-json\/wp\/v2\/posts\/507","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/ibelitetutor.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/ibelitetutor.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/ibelitetutor.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/ibelitetutor.com\/blog\/wp-json\/wp\/v2\/comments?post=507"}],"version-history":[{"count":0,"href":"http:\/\/ibelitetutor.com\/blog\/wp-json\/wp\/v2\/posts\/507\/revisions"}],"wp:attachment":[{"href":"http:\/\/ibelitetutor.com\/blog\/wp-json\/wp\/v2\/media?parent=507"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/ibelitetutor.com\/blog\/wp-json\/wp\/v2\/categories?post=507"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/ibelitetutor.com\/blog\/wp-json\/wp\/v2\/tags?post=507"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}