Calculate R8 for f(x)=8−x over [3,5].
R8=
To calculate the R8 (right-hand sum with 8 subintervals) for the function f(x) = 8 - x over the interval [3, 5], you need to follow these steps:
Step 1: Determine the width of each subinterval.
In this case, since we have 8 subintervals, we divide the interval [3, 5] into 8 equal parts. The width of each subinterval is calculated by:
Width = (Upper Limit – Lower Limit) / Number of Subintervals
= (5 - 3) / 8
= 2 / 8
= 1/4
Step 2: Calculate the right-hand sum using the given function.
For each subinterval, take the right endpoint and substitute it into the function f(x) = 8 - x. Then, multiply the result by the width of the subinterval. Finally, sum up the products for all subintervals to obtain R8.
Subinterval 1: Input = 3 + 1/4 = 13/4
f(13/4) = 8 - 13/4 = 35/4
Right term product = (35/4) * (1/4)
Subinterval 2: Input = 3 + 2/4 = 14/4
f(14/4) = 8 - 14/4 = 34/4
Right term product = (34/4) * (1/4)
Continue this process for subintervals 3 to 8.
Subinterval 3: (33/4) * (1/4)
Subinterval 4: (32/4) * (1/4)
Subinterval 5: (31/4) * (1/4)
Subinterval 6: (30/4) * (1/4)
Subinterval 7: (29/4) * (1/4)
Subinterval 8: (28/4) * (1/4)
Step 3: Calculate the sum of all the right term products.
Add up all the right term products from each subinterval to obtain R8.
R8 = (35/4)*(1/4) + (34/4)*(1/4) + (33/4)*(1/4) + (32/4)*(1/4) + (31/4)*(1/4) + (30/4)*(1/4) + (29/4)*(1/4) + (28/4)*(1/4)
Simplify the expression and calculate the final result.