Question

योग की सीमा के रूप में, `int_-1^2 (7x - 5)"d"x`  का मान निकालिए।

Answer 1

यहाँ a = -1, b = 2, तथा h = `2 + 1/"n"` है।

अर्थात्‌, nh = 3 और f (x) = 7x - 5 है।

अब, हमें प्राप्त है:

`int_(-1)^2 (7x - 5)"d"x = lim_("h" -> 0) "h"["f"(-1) + "f"(-1 + "h") + "f"(-1 + 2"h") + ... + (-1 + ("n" - 1)"h")]`

ध्यान दीजिए कि

f(–1) = –7 – 5 = –12

f(–1 + h) = –7 + 7h – 5 = –12 + 7h

f(–1 + (n –1)h) = 7 (n – 1)h – 12.

अतः, `int_(-1)^2 (7x - 5)"d"x = lim_("h" -> 0) "h"[(-12) + (7"h" - 12) + (14"h" - 12) + ... + (7("n" - 1)"h" - 12)]`

= `lim_("h" -> 0) "h"[7"h"[1 + 2 + ... +("n" - 1)] - 12"n"]`

= `lim_("h" -> 0) "h"[7"h" (("n" - 1)"n")/2 - 12 "n"]`

= `lim_("h" -> 0) [7/2("nh")("nh" - "h") - 12"nh"]`

= `7/2(3 - 0) - 12 xx 3`

= `(7 xx 9)/2 - 36`

= `(-9)/2`.