Q. 37 - Calculus

Question

For each function f and interval [a, b] in Exercises 34–38, it is possible to find the exact signed area between the graph of f and the x-axis on [a, b] geometrically by using the areas of circles, triangles, and rectangles. Find this exact area, and then calculate the left, right, midpoint, upper, lower, and trapezoid sums with n = 4. Which approximation rule is most accurate?

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

Step-by-Step Solution

Answer

The actual area of the semicircle is 1.57 square units.

Step 1. Given Information

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 

Step 2. Finding area of semicircle

The circle contains 33 full squares and 12 partial square.

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

Step 3. Left Sum

The left sum is $\sum _{k=1}^{n}f\left(x_{k-1}\right)∆x$

$=\underset{k=1}{\overset{4}{\sum (}}\sqrt{1-{\left(0.5-1.5k\right)}^2}0.5$

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

Step 4. The Right Sum

The right sum is

 $$\begin{gathered} \sum _{k=1}^{n}f\left(x_k\right)∆x=\sum _{k=1}^4\sqrt{1-{\left(0.5k-1\right)}^2}0.5 \\ =1.366 \end{gathered}$$

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

Step 5. Mid point sum

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

Step 6. Upper sum

The Upper sum is $\sum _{k=1}^{n}f\left(M_k\right)∆x$

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

Step 7. Lower Sum

The lower sum is $\sum _{k=1}^{n}f\left(m_k\right)∆x$

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

Step 8. Trapezoid sum

The trapezoid sum is given by $\sum _{k=1}^n\frac{f\left(x_{k-1}\right)+f\left(x_k\right)}{2}∆x$

$\sum _{k=1}^4\frac{\sqrt{1-{\left(0.5k-1.5\right)}^2}+\sqrt{1-{\left(0.5k-1\right)}^2}}{2}0.5=1.36$