Q.26 - Calculus

Question

Your calculator should be able to approximate the area between a graph and the x-axis. Determine how to do this on your particular calculator, and then, in Exercises 21–26, use the method to approximate the signed area between the graph of each function f and the x-axis on the given interval [a, b]. 

$f\left(x\right)=1-2^x, [a, b] = [-3, 1]$

Step-by-Step Solution

Answer

Area between the function $f\left(x\right)=1-2^x$ and x-axis on the interval [-3, 1] is $\frac{1}{8\ln \left(2\right)}+2 or 2.18$.

Step 1. Given information.

We have given function is $y=1-2^x$, x- axis and the interval is [-3, 1].

Step 2. Concept used.

Area between the curves is the area between a curve f(x) and a curve g(x) on an interval [a, b] given by,

$A={\int }_a^b\left|f\right(x)-g(x\left)\right|dx$

Step 3. Explanation.

We have $f\left(x\right)=1-2^x$, x-axis and the given interval is [-3, 1].

Area between the curve f(x) and x-axis is,

 $A={\int }_{-3}^1|1-2^x-0|dx$

      = ${\int }_{-3}^{0}1-2^{x}dx+{\int }_0^1-1+2^{x}dx$

      $$\begin{gathered} =3-\frac{7}{8\ln \left(2\right)}-1+\frac{1}{\ln \left(2\right)} \\ =\frac{1}{8\ln \left(2\right)}+2 or 2.18 \end{gathered}$$

Step 4. Conclusion.

Hence, area between the curve $f\left(x\right)=1-2^x$ and x-axis is $\frac{1}{8\ln \left(2\right)}+2 or 2.18$.