Evaluate: $\int_{1}^{4}\left(x^{2}-x\right) d x$ as limit of sums.

CBSE Class 12 · 2010 · Definite IntegralsLong Answer

Evaluate:

\int_{1}^{4} (x^{2} - x) d x

as limit of sums.

Answer: $\frac{27}{2}$

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

a = 1, b = 4, h = \frac{3}{n}, nh = 3

, with

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

\int_{1}^{4} f (x) d x = 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}]

n \to \infty

with

nh = 3

= \frac{( nh ) ^{3}}{3} + \frac{( nh ) ^{2}}{2} = \frac{27}{3} + \frac{9}{2} = 9 + \frac{9}{2} = \frac{27}{2}

CBSE Class 12 Board · 2010 · Subjective