How do you calculate the left and right Riemann sum for the given function over the interval [6,8], using n=2 for #f(x) = x^2#?
See below.
The width (length) of each subinterval will be
For the left sum, use the left endpoints and add.
For the right sum, use the right endpoints and add.
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To calculate the left and right Riemann sums for the given function ( f(x) = x^2 ) over the interval ([6,8]) using ( n = 2 ), we divide the interval ([6,8]) into ( n ) subintervals of equal width.
With ( n = 2 ), the interval ([6,8]) is divided into two subintervals:
 ([6,7])
 ([7,8])
For the left Riemann sum, we evaluate the function ( f(x) ) at the left endpoint of each subinterval and multiply it by the width of each subinterval.
For the right Riemann sum, we evaluate the function ( f(x) ) at the right endpoint of each subinterval and multiply it by the width of each subinterval.
Given that ( n = 2 ), we have:

Left Riemann sum: [ L_2 = f(6) \cdot \Delta x + f(7) \cdot \Delta x ] where ( \Delta x = \frac{{b  a}}{n} = \frac{{8  6}}{2} = 1 )

Right Riemann sum: [ R_2 = f(7) \cdot \Delta x + f(8) \cdot \Delta x ]
Now, let's compute these values:

For the left Riemann sum: [ L_2 = f(6) \cdot 1 + f(7) \cdot 1 ] [ L_2 = 6^2 \cdot 1 + 7^2 \cdot 1 ] [ L_2 = 36 + 49 ] [ L_2 = 85 ]

For the right Riemann sum: [ R_2 = f(7) \cdot 1 + f(8) \cdot 1 ] [ R_2 = 7^2 \cdot 1 + 8^2 \cdot 1 ] [ R_2 = 49 + 64 ] [ R_2 = 113 ]
Therefore, the left Riemann sum ( L_2 ) for the given function over the interval ([6,8]) using ( n = 2 ) is ( 85 ), and the right Riemann sum ( R_2 ) is ( 113 ).
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