find the area between the curve and the x - axis over the indicated interval.\ny = \\frac{6}{x}; 1, 10\nthe…

Explanation:

Step1: Recall area - under - curve formula

The area $A$ between the curve $y = f(x)$ and the $x$-axis over the interval $[a,b]$ is given by $A=\int_{a}^{b}|f(x)|dx$. Since $y=\frac{6}{x}>0$ for $x\in[1,10]$, we have $A = \int_{1}^{10}\frac{6}{x}dx$.

Step2: Integrate the function

We know that $\int\frac{1}{x}dx=\ln|x|+C$. So, $\int_{1}^{10}\frac{6}{x}dx=6\int_{1}^{10}\frac{1}{x}dx$. Using the fundamental theorem of calculus $\int_{a}^{b}F^\prime(x)dx=F(b)-F(a)$, where $F(x) = \ln|x|$ and $F^\prime(x)=\frac{1}{x}$, we get $6[\ln(x)]_{1}^{10}$.

Step3: Evaluate the definite - integral

$6[\ln(x)]_{1}^{10}=6(\ln(10)-\ln(1))$. Since $\ln(1) = 0$, the result is $6\ln(10)$.

Answer:

$6\ln(10)$