Find the area of the region bounded by the graphs of the equations $y = \frac { x + 2 } { x } , x = 1 , x = 5$, and $y = 0$. Round your answer to four decimal places.
A) $7.2189$
B) $7.5835$
C) $8.2189$
D) $8.5835$
E) $8.8283$

Question

Quizplus · Accepted Answer

To find the area of the region bounded by the given equations, we need to integrate the function $y = \frac{x + 2}{x}$ from $x = 1$ to $x = 5$, and subtract the area below the x-axis (if any, but in this case, there isn't because $y = \frac{x + 2}{x}$ is positive in the interval [1, 5]).The integral of $y = \frac{x + 2}{x}$ can be simplified to $\int (1 + \frac{2}{x}) dx = \int dx + 2\int \frac{dx}{x}$, which evaluates to $x + 2\ln|x|$ from 1 to 5.Substituting the limits, we get $(5 + 2\ln|5|) - (1 + 2\ln|1|) = 5 + 2\ln(5) - 1$, since $\ln(1) = 0$.Therefore, the area is $4 + 2\ln(5)$, which numerically approximates to $7.2189$.

Find the Area of the Region Bounded by the Graphs y=x+2x,x=1,x=5y = \frac { x + 2 } { x } , x = 1 , x = 5y=xx+2​,x=1,x=5

Find the Area of the Region Bounded by the Graphs $y = \frac { x + 2 } { x } , x = 1 , x = 5$