How do you use Riemann sums to evaluate the area under the curve of #f(x) = (e^x) − 5# on the closed interval [0,2], with n=4 rectangles using midpoints?
the answer
The sketch of our function
graph{e^x5 [16.02, 16.02, 8.01, 8.01]}
the width
The midpoints
now find the high
The sketch of our function with midpoints
calculate Riemann sum
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To use Riemann sums to evaluate the area under the curve of ( f(x) = e^x  5 ) on the closed interval ([0,2]) with ( n=4 ) rectangles using midpoints, you would follow these steps:

Determine the width of each rectangle: [ \Delta x = \frac{ba}{n} = \frac{20}{4} = 0.5 ]

Find the midpoint of each subinterval: [ x_i^* = a + \frac{(2i1)\Delta x}{2} ] where ( i = 1, 2, 3, 4 ).
For ( i = 1 ), ( x_1^* = 0 + \frac{(2 \cdot 1  1) \cdot 0.5}{2} = 0.25 )
For ( i = 2 ), ( x_2^* = 0 + \frac{(2 \cdot 2  1) \cdot 0.5}{2} = 0.75 )
For ( i = 3 ), ( x_3^* = 0 + \frac{(2 \cdot 3  1) \cdot 0.5}{2} = 1.25 )
For ( i = 4 ), ( x_4^* = 0 + \frac{(2 \cdot 4  1) \cdot 0.5}{2} = 1.75 )

Evaluate ( f(x) ) at each midpoint: [ f(x_i^) = e^{x_i^}  5 ] [ f(0.25) = e^{0.25}  5 \approx 3.942 ]
[ f(0.75) = e^{0.75}  5 \approx 2.192 ]
[ f(1.25) = e^{1.25}  5 \approx 0.745 ]
[ f(1.75) = e^{1.75}  5 \approx 0.189 ] 
Calculate the area of each rectangle: [ A_i = f(x_i^*) \cdot \Delta x ] [ A_1 = 3.942 \cdot 0.5 = 1.971 ]
[ A_2 = 2.192 \cdot 0.5 = 1.096 ]
[ A_3 = 0.745 \cdot 0.5 = 0.3725 ]
[ A_4 = 0.189 \cdot 0.5 = 0.0945 ] 
Sum up the areas of all rectangles: [ \text{Area} = A_1 + A_2 + A_3 + A_4 ] [ \text{Area} = 1.971  1.096  0.3725 + 0.0945 ] [ \text{Area} \approx 3.345 ]
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