question find the total area between the graph of the function f(x)=2 - x and the x - axis over the interval…

Explanation:

Step1: Find the x - intercept

Set $f(x)=0$, so $2 - x=0$, then $x = 2$.

Step2: Split the integral based on the x - intercept

The area $A=\int_{-3}^{2}(2 - x)dx-\int_{2}^{4}(2 - x)dx$.

Step3: Integrate $\int(2 - x)dx$

Using the power - rule of integration $\int x^n dx=\frac{x^{n + 1}}{n+1}+C(n\neq - 1)$, we have $\int(2 - x)dx=2x-\frac{x^{2}}{2}+C$.

Step4: Evaluate $\int_{-3}^{2}(2 - x)dx$

$[2x-\frac{x^{2}}{2}]_{-3}^{2}=(2\times2-\frac{2^{2}}{2})-(2\times(-3)-\frac{(-3)^{2}}{2})=(4 - 2)-(-6-\frac{9}{2})=2-(-\frac{12 + 9}{2})=2+\frac{21}{2}=\frac{4 + 21}{2}=\frac{25}{2}$.

Step5: Evaluate $\int_{2}^{4}(2 - x)dx$

$[2x-\frac{x^{2}}{2}]_{2}^{4}=(2\times4-\frac{4^{2}}{2})-(2\times2-\frac{2^{2}}{2})=(8 - 8)-(4 - 2)=0 - 2=-2$.

Step6: Calculate the total area

$A=\frac{25}{2}-(-2)=\frac{25}{2}+2=\frac{25 + 4}{2}=\frac{29}{2}$.

Answer:

$\frac{29}{2}$