Evaluate $\int_{0}^{2}\left(x^{2}+x+1\right) d x$ as limit of a sum

CBSE Class 12 · 2006 · Definite IntegralsLong Answer

Evaluate

\int_{0}^{2} (x^{2} + x + 1) d x

as limit of a sum

Answer: $\frac{20}{3}$

f (x) = x^{2} + x + 1

a = 0, b = 2, h = \frac{2}{n}, nh = 2

, with

f (r h) = r^{2} h^{2} + r h + 1

\int_{0}^{2} f (x) d x = h \to 0 lim h r = 0 \sum n - 1 (r^{2} h^{2} + r h + 1)

= h \to 0 lim [h^{3} \frac{( n - 1 ) n ( 2 n - 1 )}{6} + h^{2} \frac{( n - 1 ) n}{2} + nh]

n \to \infty

with

nh = 2

= \frac{( nh ) ^{3}}{3} + \frac{( nh ) ^{2}}{2} + nh = \frac{8}{3} + 2 + 2 = \frac{20}{3}

CBSE Class 12 Board · 2006 · Subjective