Question$\int_{1}^{4}\left(2-\frac{x}{3}\right) d x$ using Diemann Sum

Studdy Solution
Take the limit as $n$ approaches infinity:
Evaluate the sums:
$\sum_{i=1}^{n} 1 = n$
$\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$
Substitute these into the Riemann sum:
$R_n = \frac{5}{n} \cdot n - \frac{3}{n^2} \cdot \frac{n(n+1)}{2}$
Simplify:
$R_n = 5 - \frac{3(n+1)}{2n}$
Take the limit as $n \to \infty$:
$\lim_{n \to \infty} R_n = 5 - \frac{3}{2} = \frac{10}{2} - \frac{3}{2} = \frac{7}{2}$
The value of the integral is:
$\boxed{\frac{7}{2}}$