Q. 28 - 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).

$A\left(x\right)={\int }_{-\mathrm{\pi }}^x\sin t.dt$

Step-by-Step Solution

Answer

(a) The function A(2) is:

(b) The value of $A\left(2\right)=-(\cos 2+1) and A95)=-(\cos 5+1)$.

(c) The explicit elementary formula is $A\left(x\right)=-(\cos x+1)$

Part (a) Step 1. Given Information.

The function:

$A\left(x\right)={\int }_{-\mathrm{\pi }}^x\sin t.dt$

Part (a) Step 2. Graph the function.

Graph the function, $A\left(x\right)={\int }_{-\mathrm{\pi }}^x\sin t.dt$.

So graph the function $A\left(x\right)={\int }_{-\mathrm{\pi }}^x\sin t.dt$ between the points $x=-\mathrm{\pi } \mathrm{to} \mathrm{x}=2$.

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

$$\begin{gathered} A\left(x\right)={\int }_{-\mathrm{\pi }}^x\sin t.dt \\ A\left(2\right)={\int }_{-\mathrm{\pi }}^2\sin t.dt \\ ={[-\cos t]}_{-\mathrm{\pi }}^2 \\ =[-\cos 2]-[-\cos (-\mathrm{\pi }\left)\right] \\ =-(\cos 2+1) \end{gathered}$$

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

$$\begin{gathered} A\left(x\right)={\int }_{-\mathrm{\pi }}^x\sin t.dt \\ A\left(5\right)={\int }_{-\mathrm{\pi }}^5\sin t.dt \\ ={[-\cos t]}_{\mathrm{\pi }}^5 =[-\cos 5]-[-\cos \mathrm{\pi }] \\ =-(\cos 5+1) \end{gathered}$$

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

Find an explicit formula for the given function. 

$$\begin{gathered} A\left(x\right)={\int }_{-\mathrm{\pi }}^x\sin t.dt \\ ={[-\cos t]}_{-\mathrm{\pi }}^x \\ =-\cos x-\left(\cos \right(-\mathrm{\pi }\left)\right) \\ =-\cos \mathrm{x}-\cos \mathrm{\pi } \\ =-(\cos \mathrm{x}+1) \end{gathered}$$