in exercises 13 and 14, find the total area between the curve and the x - axis. 13. y = 4 - x, 0 ≤ x ≤ 6 14…

Explanation:

Step1: Find intersection with x - axis

Set $y = 0$. For $y=4 - x$, when $y = 0$, we have $4 - x=0$, so $x = 4$.

Step2: Split the integral based on sign of function

We need to split the integral of $y = 4 - x$ from $x = 0$ to $x = 6$ into two parts. The function $y=4 - x$ is non - negative on $[0,4]$ and negative on $[4,6]$. The area $A=\int_{0}^{4}(4 - x)dx-\int_{4}^{6}(4 - x)dx$.

Calculate $\int_{0}^{4}(4 - x)dx$

Using the power rule $\int x^n dx=\frac{x^{n + 1}}{n+1}+C(n\neq - 1)$ and $\int a dx=ax + C$, we have $\int_{0}^{4}(4 - x)dx=\left[4x-\frac{x^{2}}{2}\right]_{0}^{4}=(4\times4-\frac{4^{2}}{2})-(4\times0-\frac{0^{2}}{2})=16 - 8=8$.

Calculate $\int_{4}^{6}(4 - x)dx$

$\int_{4}^{6}(4 - x)dx=\left[4x-\frac{x^{2}}{2}\right]_{4}^{6}=(4\times6-\frac{6^{2}}{2})-(4\times4-\frac{4^{2}}{2})=(24 - 18)-(16 - 8)=6 - 8=-2$. So $A = 8-(-2)=10$.

Answer:

10