Question

The area bounded by the curve y = x⁴ − 2x³ + x² + 3 with x-axis and ordinates corresponding to the minima of y is
(a) 1

(b) $\frac{91}{30}$

(c) $\frac{30}{9}$

(d) 4

Answer 1

$\left(\mathrm{b}\right)\frac{91}{30}$



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