Express the area of the region bounded by the given line(s) and/or curve(s) as an iterated double integral.
-The lines $\frac { x } { 4 } + \frac { y } { 5 } = 1 , \frac { x } { 5 } + \frac { y } { 4 } = 1$, and $x = 0$
A) $\int _ { 0 } ^ { 20 / 9 } \int _ { 5 - 5 x / 4 } ^ { 4 - 4 x / 5 } d y d x$
B) $\int _ { 0 } ^ { 20 / 9 } \int _ { 4 - 4 x / 5 } ^ { 5 - 5 x / 4 } d y d x$
C) $\int _ { 0 } ^ { 9 } \int _ { 5 - 5 x / 4 } ^ { 4 - 4 x / 5 } d y d x$
D) $\int _ { 0 } ^ { 9 } \int _ { 4 - 4 x / 5 } ^ { 5 - 5 x / 4 } d y d x$

Question

Quizplus · Accepted Answer

The region is bounded by the lines $\frac { x } { 4 } + \frac { y } { 5 } = 1$ and $\frac { x } { 5 } + \frac { y } { 4 } = 1$, and by the y-axis ($x=0$).

First, we find the intersection points of the two lines:
$$\frac { x } { 4 } + \frac { y } { 5 } = 1 \ \rightarrow \ y = 5 - \frac {5}{4}x$$
$$\frac { x } { 5 } + \frac { y } { 4 } = 1 \ \rightarrow \ y = 4 - \frac {4}{5}x$$
Setting these equal to each other gives:
$$5 -\frac{5}{4}x = 4 - \frac {4}{5}x \ \rightarrow \ \frac {21}{20}x = 1 \ \rightarrow \ x = \frac{20}{21}$$
Substituting $x=\frac{20}{21}$ into either of the equations for the lines gives $y=\frac{25}{21}$.

Therefore, the region is bounded by the curves $\frac { x } { 4 } + \frac { y } { 5 } = 1$, $\frac { x } { 5 } + \frac { y } { 4 } = 1$, and $x = 0$, and it lies between the $y$-values of $4-\frac{4}{5}x$ and $5-\frac{5}{4}x$ (from left to right).

To express the area as an iterated double integral, we integrate over $x$ first, from 0 to $\frac{20}{21}$. At each value of $x$, we integrate over $y$, from the lower bound (4-$\frac{4}{5}x$) to the upper bound (5-$\frac{5}{4}x$):
$$\int _ { 0 } ^ {\frac {20}{21}} \int _ { 4 - \frac{ 4x } { 5 } } ^ { 5- \frac { 5x } { 4 } } dy\;dx$$
Simplifying yields the answer of choice B.

Express the Area of the Region Bounded by the Given x4+y5=1,x5+y4=1\frac { x } { 4 } + \frac { y } { 5 } = 1 , \frac { x } { 5 } + \frac { y } { 4 } = 14x​+5y​=1,5x​+4y​=1

Express the Area of the Region Bounded by the Given $\frac { x } { 4 } + \frac { y } { 5 } = 1 , \frac { x } { 5 } + \frac { y } { 4 } = 1$