Evaluate the following as limit of sums: $\int_{0}^{2}\left(x^{2}+x\right) d x$

CBSE Class 12 · 2005 · Definite IntegralsLong Answer

Evaluate the following as limit of sums:

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

Answer: $\frac{14}{3}$

Here

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

a = 0

b = 2

h = \frac{2}{n}

nh = 2

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

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

Writing in terms of

nh = 2

and letting

n \to \infty

= \frac{( nh ) ^{3}}{3} + \frac{( nh ) ^{2}}{2} = \frac{8}{3} + \frac{4}{2} = \frac{8}{3} + 2 = \frac{14}{3}

CBSE Class 12 Board · 2005 · Subjective