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  4. The area bounded by the curve y=(x-1)(x-2)(x-3), x-axis and ordinates x=0, x=3 is

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Q. The area bounded by the curve $y=(x-1)(x-2)(x-3), x$-axis and ordinates $x=0, x=3$ is

Application of Integrals

Solution:

image
$\left|\int\limits_0^1\left(x^3-6 x^2+11 x-6\right) d x\right|+\int\limits_1^2\left(x^3-6 x^2+11 x-6\right) d x+\left|\int\limits_2^3\left(x^3-6 x^2+11 x-6\right) d x\right| $
$\left.\left.\left.\mid \frac{x^4}{4}-2 x^3+\frac{11}{2} x^2-6 x\right]_0^1 \mid+\frac{x^4}{4}-2 x^3+\frac{11}{2} x^2-6 x\right]_1^2+\mid \frac{x^4}{4}-2 x^3+\frac{11}{2} x^2-6 x\right]_2^3 \mid$
$\left|\frac{23}{4}-8\right|+\left(-2-\left(-\frac{9}{4}\right)\right)+\left|-\frac{9}{4}+2\right|=\frac{11}{4}$