Q.Find the area of the region bounded by the line $x=2y$, the y-axis, and the horizontal lines $y=1$ and $y=4$.

The region is walled by the y-axis and capped by horizontal lines, so integrate with respect to $y$ using $A=\int x\,dy$ with $x=2y$.

Step 1 — choose the formula. The region is bounded by the y-axis and the horizontal lines $y=1$ and $y=4$, so use horizontal strips:

Step 2 — substitute $x$ in terms of $y$. The curve is given as $x=2y$, so

Step 3 — integrate and apply the limits. …

Dual-check by the trapezium (geometry). At $y=1$, the horizontal distance from the y-axis to the line is $x=2(1)=2$; at $y=4$, it is $x=2(4)=8$. The region is a trapezium with parallel sides $2$ and $8$ and height $4- …

Integrating with respect to $x$ instead of $y$. Because the bounding lines are horizontal and the region rests against the y-axis, the cor …