91影视

The given function is $f\left(x\right)=\sqrt{1-x^2},\left[a,b\right]=\left[-1,1\right]$

The graph of the function shows semicircle

The circle contains 33 full squares and 12 partial square.

So area is$33+\frac{1}{2}脳12=39$square unit

The left sum is $\underset{k=1}{\overset{n}{鈭�}}f\left(x_{k-1}\right)鈭�x$

role="math" localid="1649607934154" $=\underset{k=1}{\overset{4}{鈭�(}}\sqrt{1-{\left(0.5-1.5k\right)}^2}0.5$

The left sum is not possible because it has negative sign inside square roots.

The right sum is

$$\begin{gathered} \underset{k=1}{\overset{n}{鈭�}}f\left(x_k\right)鈭�x=\underset{k=1}{\overset{4}{鈭�}}\sqrt{1-{\left(0.5k-1\right)}^2}0.5 \\ =1.366 \end{gathered}$$

$$\begin{gathered} \underset{k=1}{\overset{4}{鈭�}}\sqrt{1-{\left(0.5k-1\right)}^2}\left(0.5\right) \\ =0.468+0.5+0.43+0=1.39 \end{gathered}$$

$$\begin{gathered} \underset{k=1}{\overset{n}{鈭�}}f\left(\frac{x_{k-1}+x_k}{2}\right)鈭�x=\underset{k=1}{\overset{4}{鈭�}}\left(\sqrt{1-{\left(\frac{k-2.5}{2}\right)}^2}\right)0.5 \\ =0.330+0.48+0.48+0.33=1.62 \end{gathered}$$

The Upper sum is $\underset{k=1}{\overset{n}{鈭�}}f\left(M_k\right)鈭�x$

$$\begin{gathered} f\left(-0.5\right)鈭�x+f\left(0\right)鈭�x+f\left(0.5\right)鈭�x+f\left(1\right)鈭�x \\ =0.86脳1+1脳1+0.86脳1+0脳1=2.72 \end{gathered}$$

The lower sum is $\underset{k=1}{\overset{n}{鈭�}}f\left(m_k\right)鈭�x$

$$\begin{gathered} f\left(-1\right)鈭�x+f\left(-0.5\right)鈭�x+f\left(0\right)鈭�x+f\left(0.5\right)鈭�x \\ =0+0.86+1+0.86=2.72 \end{gathered}$$

The trapezoid sum is given by $\underset{k=1}{\overset{n}{鈭�}}\frac{f\left(x_{k-1}\right)+f\left(x_k\right)}{2}鈭�x$

$\underset{k=1}{\overset{4}{鈭�}}\frac{\sqrt{1-{\left(0.5k-1.5\right)}^2}+\sqrt{1-{\left(0.5k-1\right)}^2}}{2}0.5=1.36$