Evaluate ∫ (x + 1/2) ; x ∈ (0,5) dx as a limit of a sum.

Question

Evaluate \(\int^{5}_{0}\left(x+\frac{1}{2}\right)\)dx as a limit of a sum.

Answer 1

We have,

\(\int^ b_a f(x)dx=\lim\limits_{x \to 0}\,h\,[f(a)+f(a+h)+f(a+2h)+...\\+f(a+(n-1)h)]\)

Where h = \(\frac{b-a}{n}\)

Here,

\(a=0,b=5,f(x)=x+\frac{1}{2}\) and

\(h=\frac{5-0}{n}=\frac{5}{n}\)

\(I=\lim\limits_{x \to 0}\,\frac{5}{n}\left[\frac{n}{2}+\frac{5}{2}(n-1)\right]\) \(\left[\therefore\,h=\frac{5}{n}\,and \,h \to 0⇒ n \to \infty \right]\)