Evaluate the following integral as a limit of sums: ∫ (3x^2 + 1)dx, x ∈ [1, 3]

Question

Evaluate the following integral as a limit of sums:

\(\int\limits_{1}^{3} \) (3x² + 1)dx

Answer 1

To find: \(\int\limits_{1}^{3} \) (3x² + 1)dx

Formula used:

where,

Here, f(x) = 3x² + 1 and a = 1

Now, by putting x = 1 in f(x) we get,

f(1) = 3(1²) + 1 = 3(1) + 1 = 3 + 1 = 4

f(1 + h)

= 3(1 + h)² + 1 = 3{h² + 1² + 2(h)(1)} + 1

= 3(h)² + 3 + 3(2h) + 1

= 3(h)² + 4 + 6h

Similarly, f(1 + 2h)

= 3(1 + 2h)² + 1

= 3{2(2h)² + 1² + 2(2h)(1)} + 1

= 3(2h)² + 3 + 3(4h) + 1

= 3(2h)² + 4 + 12h

{∵ (x + y)² = x² + y² + 2xy}

Put,

h = \(\cfrac2n\)

Since,

⇒ I = 28

Hence, the value of