Q. 65 - Calculus

Question

For each function f and interval [a, b] in Exercises 56–67, use definite integrals and the Fundamental Theorem of Calculus to find the exact average value of f from x = a to x = b. Then use a graph of f to verify that your answer is reasonable.  

$f\left(x\right)=2-\sqrt[3]{x}, [a, b] = [1, 8]$

Step-by-Step Solution

Answer

The exact average value of f  is$0.39,$and it is verified from the graph of f.

The graph is

Step 1. Given Information.

The given function and interval is $f\left(x\right)=2-\sqrt[3]{x}, [a, b] = [1, 8].$

Step 2. Finding the exact average value.

To find the exact average value of f  from $x=a to x=b,$ we will use the formula: $\frac{1}{b-a}{\int }_a^{b}f\left(x\right) dx.$

Thus,

$$\begin{gathered} \frac{1}{b-a}{\int }_a^{b}f\left(x\right) dx \\ =\frac{1}{8-1}{\int }_1^{8}2-\sqrt[3]{x} dx \\ =\frac{1}{7}\left\{{\int }_1^{8}2dx-{\int }_1^8\sqrt[3]{x} dx\right\} \\ =\frac{1}{7}\left\{2{\int }_1^{8}dx-{\int }_1^8\sqrt[3]{x} dx\right\} \\ =\frac{1}{7}\left\{2{\left[x\right]}_1^8-{\left[\frac{x^{{\displaystyle \raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}}{{\displaystyle \frac{4}{3}}}\right]}_1^8\right\} \\ =\frac{1}{7}\left\{2\left[8-1\right]-\left[\frac{3}{4}\left(8^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}-1^{\raisebox{1ex}{$8$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right)\right]\right\} \\ =\frac{1}{7}\left\{14-\left(\frac{3}{4}\left(16-1\right)\right)\right\} \\ =\frac{1}{7}\left(14-\frac{45}{4}\right) \\ =\frac{1}{7}\left(\frac{56-45}{4}\right) \\ =\frac{1}{7}\left(\frac{11}{4}\right) \\ =\frac{11}{28} \\ \approx 0.39 \end{gathered}$$

Step 3. Verification.

By using the graphing utility the graph of f  is 

From the graph, we can depict that the average value is $0.39.$ Thus, the answer is right.