Correct Answer - Option 1 : 104 sq. units
CONCEPT:
- \(\smallint {{\rm{x}}^{\rm{n}}}{\rm{dx}} = \frac{{{{\rm{x}}^{{\rm{n}} + 1}}}}{{{\rm{n}} + 1}} + {\rm{C}}\)
CALCULATION:
Here, we have to find the area of the region bounded by the curves y = x3 + 4x + 2, the line x = 2, x = 5 and the x - axis
So, the area enclosed by the given curves is given by \(\mathop \smallint \nolimits_0^4 {(x^3 + 4x + 2)}\;dx\)
As we know that, \(\smallint {{\rm{x}}^{\rm{n}}}{\rm{dx}} = \frac{{{{\rm{x}}^{{\rm{n}} + 1}}}}{{{\rm{n}} + 1}} + {\rm{C}}\)
\(⇒ \mathop \smallint \nolimits_0^4 {(x^3 + 4x + 2)}\;dx = \left[ {\frac{{{x^4}}}{4} +\frac{4x^2}{2} + 2x} \right]_0^4\)
⇒ \(\frac{1}{4}\;\left( {256) + \frac{64}{2} + (2 \times4\;} \right) = 104 \;sq.\;units\)
Hence, option A is the correct answer.
