Which inequality has a solid boundary line when graphed?
1 answer:
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Partitioning the interval ![[6,11]](https://tex.z-dn.net/?f=%5B6%2C11%5D)
into 
equally-spaced subintervals gives rectangles of width 
and of heights determined by the right endpoints of each subinterval.
If
, then 
, 
, and so on, up to 
. Because we're using the right endpoints, the approximation will consider 
The definite integral is then approximated by
You have
To check that this is correct, let's make sure the sum converges to the exact value of the definite integral. As
, you have the sum converging to 
.
Meanwhile,
![\int_6^{11}(1-5x)\,\mathrm dx=\left[x-\dfrac52x^2\right]_{x=6}^{x=11}=-\dfrac{415}2](https://tex.z-dn.net/?f=%5Cint_6%5E%7B11%7D%281-5x%29%5C%2C%5Cmathrm%20dx%3D%5Cleft%5Bx-%5Cdfrac52x%5E2%5Cright%5D_%7Bx%3D6%7D%5E%7Bx%3D11%7D%3D-%5Cdfrac%7B415%7D2)
so we're done.
If
The definite integral is then approximated by
You have
To check that this is correct, let's make sure the sum converges to the exact value of the definite integral. As
Meanwhile,
so we're done.
