Question

योग की सीमा के रूप में निम्नलिखित निश्चित समाकलन का मान ज्ञात कीजिए।

`int_1^4 ("x"^2 - "x")` dx

Answer 1

यहाँ f(x) = x² - 2 a = 1, b = 4

तब nh = b - a = 3

परिभाषानुसार,

`int_a^b "f"("x") "dx" = lim_("h" -> 0)` h [f(x) + f(a + h) + f(a + 2h)  .... f(a + (n - 1) h)]

जहाँ nh = 3

अब f(a) = f(1) = 1² - 1 = 0

f(a + h) = f(1 + h) = (1 + h)² - (1 + h) = h² + h

f(a + 2h) = f(1 + 2h) = (1 + 2h)² - (1 + 1h) = (2h)² + 2h

f{a = (n - 1) h} = f {1 + (n - 1) h} = [1 + (n - 1) h]² - [1 + (n - 1)h] { (n - 1)h}² + (n - 1)h 

`therefore int_1^4  ("x"^2 - "x") "dx" =  lim_("h" -> 0)` h [0 + (h² + h) + {(2h)² + 2h) .... + [(n - 1)² h² + (n - 1) h}]

`=  lim_("h" -> 0)` [{1² + 2² + 3² .... (h - 1)²} + h {1 + 2 + 3 .... (n - 1)²}]

`=  lim_("h" -> 0) ["h"^2 (("n" - 1)*"n"*(2"n" - 1))/6 + "h" (("n" + 1)"h")/2]`

`=  lim_("h" -> 0) [(("nh - h") "nh" (2"nh - h"))/6 + (("nh - h")"nh")/2]`

`=  lim_("h" -> 0) [((3 - "h")3* (2 xx 3 - "h"))/6 + ((3 - "h")*3)/2]`

`= (3 xx 3 xx 6)/6 + (3 xx 3)/2`

`= 9 + 9/2`

`= 27/2`