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

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KollegeApply · Accepted Answer

$\frac{91}{30}$ 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[ x^$\frac{5}{5}$ - \frac{2 \left( x \right)^4}{4} + x^$\frac{3}{3}$ + 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 } [Solution diagram: 44008_solution_1.png]

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