The area of the region {(x,y) : |x-y| <= 4sqrt(x)} is

The correct answer is Option 2. Full mathematical solution steps and hints are accessible in the interactive StemCET workspace.

Maths Q19 – JEE-MAIN 2025

← Back

Home Maths IntegralsQuestion #219

The area of the region

{(x, y) : | x - y | \leq 4 \sqrt{x}}

Clue 1 of 3: Interpret the inequality

The absolute value inequality $| A | \leq B$ can be rewritten as $- B \leq A \leq B$ .

🧠 Full Solution Path

Step 1: Rewrite the inequality to define the region✦ Active

The given inequality is $| x - y | \leq 4 \sqrt{x}$ . For $\sqrt{x}$ to be defined, we must have $x \geq 0$ . The absolute value inequality can be written as:

- 4 \sqrt{x} \leq x - y \leq 4 \sqrt{x}

This gives two inequalities for $y$ :

x - y \leq 4 \sqrt{x} ⟹ y \geq x - 4 \sqrt{x}

x - y \geq - 4 \sqrt{x} ⟹ y \leq x + 4 \sqrt{x}

So, the region is bounded by the curves $y_{1} = x - 4 \sqrt{x}$ (lower bound) and $y_{2} = x + 4 \sqrt{x}$ (upper bound).

Step 2: Determine the limits of integration○ Expand

To find the limits of integration, we need to find where the lower curve $y_{1} = x - 4 \sqrt{x}$ intersects the x-axis (i.e., $y_{1} = 0$ ):

x - 4 \sqrt{x} = 0

\sqrt{x} (\sqrt{x} - 4) = 0

This yields $\sqrt{x} = 0 ⟹ x = 0$ or $\sqrt{x} = 4 ⟹ x = 16$ . These are the limits of integration.

💡 Teacher's Secret Hint

The region is enclosed between the two curves from their intersection points. In this case, the lower curve intersects the x-axis at the points that define the extent of the region.

Step 3: Calculate the area by integration○ Expand

The area $A$ of the region is given by the integral of the difference between the upper and lower curves from $x = 0$ to $x = 16$ :

A = \int_{0}^{16} (y_{2} - y_{1}) d x

A = \int_{0}^{16} ((x + 4 \sqrt{x}) - (x - 4 \sqrt{x})) d x

A = \int_{0}^{16} (8 \sqrt{x}) d x

A = 8 \int_{0}^{16} x^{1 / 2} d x

A = 8 {[\frac{x^{3 / 2}}{3 / 2}]}_{0}^{16}

A = 8 {[\frac{2}{3} x^{3 / 2}]}_{0}^{16}

A = \frac{16}{3} [(16)^{3 / 2} - (0)^{3 / 2}]

A = \frac{16}{3} [(\sqrt{16})^{3} - 0]

A = \frac{16}{3} [4^{3}]

A = \frac{16}{3} [64]

A = \frac{1024}{3}

💡 Teacher's Secret Hint

Remember to correctly evaluate the power $x^{3 / 2}$ as $(\sqrt{x})^{3}$ or $(x^{1 / 2})^{3}$ .

✦ STEM Console utilizes AI models to generate step-by-step explanations and math clues. AI can make mistakes.

Maths Question 19 – JEE-MAIN 2025

Clue 1 of 3: Interpret the inequality

🧠 Full Solution Path