Evaluate: $\int_{1}^{3}\left(3 x^{2}+2 x\right) d x$ as limit of sums.

CBSE Class 12 · 2010 · Definite IntegralsLong Answer

Evaluate:

\int_{1}^{3} (3 x^{2} + 2 x) d x

as limit of sums.

Answer: $34$

f (x) = 3 x^{2} + 2 x

a = 1, b = 3, h = \frac{2}{n}, nh = 2

, with

f (1 + r h) = 3 (1 + r h)^{2} + 2 (1 + r h) = 5 + 8 r h + 3 r^{2} h^{2}

\int_{1}^{3} f (x) d x = h \to 0 lim h r = 0 \sum n - 1 (5 + 8 r h + 3 r^{2} h^{2})

= h \to 0 lim [5 nh + 8 h^{2} \frac{( n - 1 ) n}{2} + 3 h^{3} \frac{( n - 1 ) n ( 2 n - 1 )}{6}]

n \to \infty

with

nh = 2

= 5 (nh) + 4 (nh)^{2} + (nh)^{3} = 5 (2) + 4 (4) + 8 = 10 + 16 + 8 = 34

CBSE Class 12 Board · 2010 · Subjective