1. Identify the curves inside the square

The square region is bounded by: $x=0,\; y=0,\; x=4,\; y=4$ so it is the square $[0,4]\times[0,4]$.

The given parabolas are: $y^2=4x \quad \Rightarrow \quad x=\frac{y^2}{4}$ $x^2=4y \quad \Rightarrow \quad y=\frac{x^2}{4}$

Both lie in the first quadrant.

2. Find intersection points of the two parabolas

Solve $y^2=4x, \qquad x^2=4y$

From $x=\dfrac{y^2}{4}$, substitute into $x^2=4y$: $\left(\frac{y^2}{4}\right)^2=4y$ $\frac{y^4}{16}=4y$ $y^4=64y$ $y(y^3-64)=0$ So, $y=0 \quad \text{or} \quad y=4$ Hence corresponding $x$ values are $x=0$ and $x=4$.

Thus the curves intersect at: $(0,0), \; (4,4)$

3. Understand the three regions

Inside the square, the two curves divide it into three horizontal parts from top to bottom:

• Top region $S_1$: above $y=\dfrac{x^2}{4}$
• Middle region $S_2$: between the two curves
• Bottom region $S_3$: below $x=\dfrac{y^2}{4}$, i.e. below $y=2\sqrt{x}$

It is easiest to compute using integration with respect to $x$.

For $0\le x\le 4$:

• upper parabola (opening right) gives $y=2\sqrt{x}$
• lower parabola (opening upward) gives $y=\frac{x^2}{4}$

Also note that on $[0,4]$, $2\sqrt{x} \ge \frac{x^2}{4}$ so the middle strip is between these two curves.

4. Area of middle region $S_2$

$S_2=\int_0^4 \left(2\sqrt{x}-\frac{x^2}{4}\right)dx$

Compute separately: $\int_0^4 2\sqrt{x}\,dx=2\int_0^4 x^{1/2}dx=2\cdot \frac{2}{3}x^{3/2}\Big|_0^4$ $=\frac{4}{3}\cdot 8=\frac{32}{3}$

$\int_0^4 \frac{x^2}{4}dx=\frac14\cdot \frac{x^3}{3}\Big|_0^4=\frac14\cdot \frac{64}{3}=\frac{16}{3}$

Therefore, $S_2=\frac{32}{3}-\frac{16}{3}=\frac{16}{3}$

5. Area of top region $S_1$

This is the area between $y=4$ and $y=2\sqrt{x}$: $S_1=\int_0^4 (4-2\sqrt{x})dx$

$\int_0^4 4\,dx=16$ $\int_0^4 2\sqrt{x}\,dx=\frac{32}{3}$

So, $S_1=16-\frac{32}{3}=\frac{16}{3}$

6. Area of bottom region $S_3$

This is the area between $y=0$ and $y=\dfrac{x^2}{4}$: $S_3=\int_0^4 \frac{x^2}{4}dx=\frac{16}{3}$

7. Compare the three areas

$S_1=S_2=S_3=\frac{16}{3}$

Hence, $S_1:S_2:S_3=1:1:1$

8. Check options

• A: $1:2:1$ ❌
• B: $1:2:3$ ❌
• C: $2:1:2$ ❌
• D: $1:1:1$ ✅

So the correct option is D.