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Find an upper bound for definite integral sqrt(x) *dx on [1,6] by calculating the upper...

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cleowen | Student | eNoter

Posted July 31, 2012 at 4:33 AM via web

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Find an upper bound for definite integral sqrt(x) *dx on [1,6] by calculating the upper Riemann sum, using the partition x0=1, x1=3, x2=6 of the interval [1,6].



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embizze | High School Teacher | (Level 1) Educator Emeritus

Posted July 31, 2012 at 5:09 AM (Answer #1)

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Find the approximation for `int_1^6sqrt(x)dx` on [1,6] using Riemann upper sum and two rectangles; the endpoints of the rectangles are x=1,3,and 6.

The Riemann upper sum is given by:

`sum_(k=1)^nf(x_i^("*"))Deltax_i` ; since the function is monotonically increasing we use the right endpoint of each rectangle. We have `Deltax_1=2` and `Delta x_2=3`

The approximation is given by:

`sum[sqrt(3)*2+sqrt(6)*3]~~10.813`

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The approximation for `int_1^6sqrt(x)dx` using the given rectangles is 10.813

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Compare to the actual value of approximately 9.1312923 -- for a monotonically increasing function, the Riemann upper sum gives an overestimation.

The graph:

Sources:

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