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

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How do you use Riemann sums to evaluate the area under the curve of $f (x) = (e^{x}) - 5$ $f(x) = (e^x) − 5$ 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
$S_{p} = - 3.67701446661601$ $S_p=-3.67701446661601$

Explanation:

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

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

the width

$w i d t h = \frac{2 - 0}{4} = \frac{1}{2}$ $width=(2-0)/4=1/2$

The midpoints

$\frac{0 + \frac{1}{2}}{2} = \frac{1}{4}$ $(0+1/2)/2=1/4$
$\frac{\frac{1}{2} + 1}{2} = \frac{3}{4}$ $(1/2+1)/2=3/4$
$\frac{1 + \frac{3}{2}}{2} = \frac{5}{4}$ $(1+3/2)/2=5/4$
$\frac{\frac{3}{2} + 2}{2} = \frac{7}{4}$ $(3/2+2)/2=7/4$

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

The sketch of our function with midpoints

enter image source here

calculate Riemann sum

$S_{p} = w i d t h \cdot h i g h$ $S_p=width*high$

$S_{p} = (\frac{1}{2}) [- 3.71597458331226 - 2.88299998338733 - 1.50965704253816 + 0.75460267600573]$ $S_p=(1/2)[-3.71597458331226-2.88299998338733-1.50965704253816+0.75460267600573]$

$S_{p} = - 3.67701446661601$ $S_p=-3.67701446661601$

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

1 Answer

Explanation:

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

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Explanation:

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