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?

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Answer 1 · 2018-05-06T00:03:02

the answer
$S_p=-3.67701446661601$

The sketch of our function $f(x) = (e^x) − 5$

graph{e^x-5 [-16.02, 16.02, -8.01, 8.01]}

the width

$width=(2-0)/4=1/2$

The midpoints

$(0+1/2)/2=1/4$
$(1/2+1)/2=3/4$
$(1+3/2)/2=5/4$
$(3/2+2)/2=7/4$

now find the high
$f(1/4)=-3.71597458331226$
$f(3/4)=-2.88299998338733$
$f(5/4)=-1.50965704253816$
$f(7/4)=0.75460267600573$

The sketch of our function with midpoints

enter image source here

calculate Riemann sum

$S_p=width*high$

$S_p=(1/2)[-3.71597458331226-2.88299998338733-1.50965704253816+0.75460267600573]$

$S_p=-3.67701446661601$

How do you use Riemann sums to evaluate the area under the curve of f(x) = (e^x) − 5f(x) = (e^x) − 5 on the closed interval [0,2], with n=4 rectangles using midpoints?