Q. 29 - Calculus

Question

For each area accumulation function A in Exercises 27–30, 

(a) illustrate A(2) graphically, 

(b) calculate A(2) and A(5), and 

(c) find an explicit elementary formula for A(x).

$A\left(x\right)={\int }_3^x\frac{1}{t}.dt$

Step-by-Step Solution

Answer

(a) The function A(2) is:

(b) $$\begin{gathered} A\left(2\right)=\ln 2-\ln 3 \\ A\left(5\right)=\ln 5-\ln 3 \end{gathered}$$

(c) The explicit elementary formula is $A\left(x\right)=\ln x-\ln 3$

Part (a) Step 1. Given Information.

The function:

$A\left(x\right)={\int }_3^x\frac{1}{t}.dt$

Part (a) Step 2. Graph the function.

Graph the function, 

So graph the function $A\left(2\right)={\int }_3^2\frac{1}{t}.dt$ between the points $x=2, x=3$.

Part (b) Step 1. Calculate A(2).

$$\begin{gathered} A\left(x\right)={\int }_3^x\frac{1}{t}.dt \\ A\left(2\right)={\int }_3^2\frac{1}{t}.dt \\ ={[\ln t]}_3^2 \\ =\ln 2-\ln 3 \end{gathered}$$

Part (b) Step 2. Calculate A(5).

$$\begin{gathered} A\left(5\right)={\int }_3^5\frac{1}{t}.dt \\ =[\ln t{\}}_3^5 \\ =\ln 5-\ln 3 \end{gathered}$$

Part (c) Step 1. Find an explicit formula.

Find an explicit formula for the given function. 

$$\begin{gathered} A\left(x\right)={\int }_3^x\frac{1}{t}.dt \\ ={[\ln t]}_3^x \\ =\ln x-\ln 3 \end{gathered}$$