Question

Find the area of the region bounded by the $y$-axis, the line $y=6$, and the line $y = \frac{1}{2}x$.

Answer 1

For a better understanding of the solution provided here please go through the diagram in the file that has been attached.

As can be clearly seen from the question and the graph, the area is bounded by a straight line $y=\frac{1}{2}x$ which passes through the origin, the y axis and the horizontal line $y=6$.

As we can see from the graph, this is a right triangle. To find the point of intersection of the horizontal line y=6 and the line $y=\frac{x\1}{2}x$, we will have to equate the two as:

$\frac{1}{2}x=6$

Therefore, $x=12$

Thus, $y=\frac{1}{2}x= \frac{1}{2}\times 12=6$

Hence the point of intersection is (12,6) which is depicted on the graph too.

As this is a case of a simple right triangle, the area of the bounded region will thus be:

$Area=\frac{1}{2}\times base\times height$

Let the base be the y=6 line whose length is 12 and let height be the y axis whose length is 6, thus, the area will be:

$Area=\frac{1}{2}\times 6\times 12=36$ squared units.