Q.Find the area of the region bounded by the curve $x=y^{2}$, the y-axis, and the horizontal lines $y=0$ and $y=3$.

The region is bounded by the y-axis and two horizontal lines, so integrate with respect to $y$ using $A=\int x\,dy$ with $x=y^{2}$.

Step 1 — choose the right formula. The region is bounded by the curve, the y-axis, and the horizontal lines $y=0$ and $y=3$, so the area is measured with horizontal strips:

Step 2 — express $x$ in terms of $y$. The curve is already given as $x=y^{2}$, so

$A=\int_{0}^{3} y^{2}\,dy$ …

Dual-check by the complementary region. The full rectangle from $x=0$ to $x=9$ and $y=0$ to $y=3$ has area $9\times3=27$. The area to the LEFT of the curve (between the curve and the y-axis, computed above) plus the area BELOW the curve (between the curve and the x-axis) must fill this rectangle. The area below the curve is …

Integrating with respect to $x$ out of habit. Because the region is walled by the y-axis and capped by horizontal lines, the strips are horizontal and the …