Evaluate the definite integral int_0^4 x[x] , | vedclass

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Question

(A) Let $I = \int_0^4 x[x] \, dx$. Since $[x]$ is a step function,we split the integral at integer points: $I = \int_0^1 x[0] \, dx + \int_1^2 x[1] \,

Vedclass · Accepted Answer

$17$

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(A) Let $I = \int_0^4 x[x] \, dx$. Since $[x]$ is a step function,we split the integral at integer points: $I = \int_0^1 x[0] \, dx + \int_1^2 x[1] \, dx + \int_2^3 x[2] \, dx + \int_3^4 x[3] \, dx$. $I = 0 + \int_1^2 x \, dx + \int_2^3 2x \, dx + \int_3^4 3x \, dx$. $I = \left[ \frac{x^2}{2} \right]_1^2 + 2 \left[ \frac{x^2}{2} \right]_2^3 + 3 \left[ \frac{x^2}{2} \right]_3^4$. $I = \left( \frac{4-1}{2} \right) + (9-4) + \frac{3}{2}(16-9)$. $I = \frac{3}{2} + 5 + \frac{21}{2}$. $I = \frac{3+21}{2} + 5 = \frac{24}{2} + 5 = 12 + 5 = 17$.

Evaluate the definite integral $\int_0^4 x[x] \, dx$,where $[x]$ denotes the greatest integer function not greater than $x$.

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