Question

The area bounded by the curve y = x⁴ − 2x³ + x² + 3 with x-axis and ordinates corresponding to the minima of y is _________ .

Options

• 1

• \[\frac{91}{30}\]

• \[\frac{30}{9}\]

• 4

Answer 1

\[\frac{91}{30}\]

 

Clearly, from the figure the minimum value of y is 3 when x = 0 or 1.
Therefore, the required area ABCD,
\[A = \int_0^1 y d x ............\left(\text{Where, } y = x^4 - 2 x^3 + x^2 + 3 \right)\]
\[ = \int_0^1 \left( x^4 - 2 x^3 + x^2 + 3 \right) d x\]
\[ = \left[ \frac{x^5}{5} - \frac{2 \left( x \right)^4}{4} + \frac{x^3}{3} + 3x \right]_0^1 \]
\[ = \left[ \frac{\left( 1 \right)^5}{5} - \frac{2 \left( 1 \right)^4}{4} + \frac{\left( 1 \right)^3}{3} + 3\left( 1 \right) \right] - \left[ \frac{\left( 0 \right)^5}{5} - \frac{2 \left( 0 \right)^4}{4} + \frac{\left( 0 \right)^3}{3} + 3\left( 0 \right) \right]\]
\[ = \left[ \frac{1}{5} - \frac{1}{2} + \frac{1}{3} + 3 \right] - 0\]
\[ = \frac{6 - 15 + 10 + 90}{30}\]
\[ = \frac{91}{30}\text{ square units }\]