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int_(a)^(b) x dx |
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Answer» Solution :we know that `overset(b)UNDERSET(a)(int) f(x) dx= underset( h TO0)(" LIM") h [f(a)+f(a+h)` `+f(a+2h)+....+f{a+(n-1)h}]` ` " where " nh=b-a` ` " Here " a=a,b= b " and" f(x)=x` ` :.underset(a)overset(b)(int) x dx =underset( h to 0)("lim") h[a+(a+h)+(a+2h)` `+.....+a+(n-1)h]` `underset(h to0)("lim") h[(a+a+.....+ n " TIMES ")` `+h{1+2+.....(n-1)}]` `=underset(h to 0)("lim") [hna +h^(2).(n(-1))/(2)]` `=underset(h to 0)("lim") [hna +.(hn(nh-h))/(2)]` `=underset(h to 0)("lim")[(b-a)a+.((b-a)(b-a-h))/(2)]"(":. nh =b-a")"` `=[(b-a)a+.((b-a)^(2))/(2)]` `=(b-a)(a+(b-1)/(2))=(b-a)((2a+b-a)/(2))` `=(b-a)((a+b)/(2))=(b^(2)-a^(2))/(2)` |
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