Question

Theorem

Answer 1

if f is integrable on [a,b], then
\[\int\limits_{a}^{b} f(x)dx=\lim_{n \rightarrow \infty} \sum_{j=1}^{n}f(x_j)Deltax\]
where \[\Delta(x)=\frac{ b-a }{ n } and x_j=a+j \Delta x\]
evaluate the integral using the above theorem.

Answer 2

\[\int\limits_{a}^{b} f(x)dx=\lim_{n \rightarrow \infty} \sum_{j=1}^{n}f(x_j)\Delta x\]
Where \(\Delta(x)=\frac{ b-a }{ n } \) and \(x_j=a+j \Delta x\)


\[\qquad\qquad =\lim_{n \rightarrow \infty} \sum_{j=1}^{n}f\left(a+j \frac{ b-a }{ n }\right)\frac{ b-a }{ n } \]