InterviewSolution
Saved Bookmarks
InterviewSolution
| 1. |
योगफल की सिमा के रूप में `int_(0)^(2) 2^(x)dx` का मान ज्ञात कीजिएः। |
|
Answer» यहाँ `a=0,b=2,f(x)=2^(x), nh=b-a=2 ` योगफल की सिमा की परिभाषा से, `underset(a)overset(b)int f(x)dx=underset(h to 0)lim h[f(a)+f(a+h)+f(a+2h)+.....+f{a+(n-1)h}]` `underset(0)overset(2)int 2^(x)dx` `=underset(h to 0)limh[f(0)+f(h)+f(2h)+...f{(n-1)h}]` `=underset(h to 0)lim h[2^(0)+2^(h)+2^(2h)+...2^((n-1))]` `=underset( to 0)lim h[1+2^(h)+2^(2h)+...2^((n-1)h)]` `=underset( to 0)lim {{(2h)^(n)-1)/((2^(h)-1))} [therefore a+ar+...+ar^(n-1)=(a(r^(n-)1))/(r-1)` `=underset(h to 0)lim h((2^(nh)-1))/((2^(h)-1))=underset(h to 0)lim ((2^(2)-1))/(((2^(h)-1)/(n)))` `=((2^2-1))/(log_(e)2)=(3)/(log_(e)2), [therefore underset(h to 0)lim (2^(h)-1)/(h)=log2]` |
|