The odd natural number a, such that the area of the region bounded by y = 1, y = 3, x = 0, x = y^a i

Q: Why do we integrate with respect to $y$ instead of $x$?

Great question! In calculus, we choose the variable of integration based on the boundaries. Here, the boundaries are horizontal lines ($y=1$ and $y=3$). If we integrated with respect to $x$, we would have to invert the function $x=y^a$ to $y=x^{1/a}$ and deal with complex limits. Integrating with respect to $y$ keeps the limits as simple constants, making the calculation much cleaner.

Q: Can we solve $\frac{3^{a+1}-1}{a+1} = \frac{364}{3}$ algebraically?

While it is theoretically possible to use numerical methods or logarithms, it is not practical for a JEE Advanced exam. The equation involves both an exponential term $3^{a+1}$ and a linear term $a+1$ in the denominator. This is a classic 'trap' designed to test your time-management skills. Testing the given options is the intended, most efficient strategy.

Q: What if $a$ was not an odd natural number?

The problem specifies $a$ is an odd natural number to ensure the function $x=y^a$ is well-behaved and monotonic in the interval $[1, 3]$. If $a$ were a fraction or negative, the shape of the curve would change, potentially affecting the area calculation. Always pay attention to the domain constraints provided in the question!

Question

The odd natural number $a$, such that the area of the region bounded by $y = 1, y = 3, x = 0, x = y^a$ is $\frac{364}{3}$, equal to:

Options:

(A) 3

(B) 5

(C) 7

(D) 9

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing on the Cartesian plane. You have two horizontal lines, $y=1$ and $y=3$, acting as your floor and ceiling.

To your left, you have the $y$-axis, defined by $x=0$. To your right, you have a mysterious curve, $x=y^a$, where $a$ is an odd natural number. This is the region we are tasked with measuring.

### The Power of Integration

When we face an area problem, the first question we must ask is: "Which variable should I integrate with respect to?" Since our boundaries are horizontal lines ($y=1$ and $y=3$), the most elegant approach is to integrate with respect to $y$.

The area $A$ is given by the integral of the width $x$ over the height $dy$. Thus, our formula is:

$$A = \int_{1}^{3} x \, dy$$

Substituting our curve $x=y^a$, we get the integral:

$$A = \int_{1}^{3} y^a \, dy$$

### The Integration Journey

Now, let us perform the integration. Using the power rule, $\int y^n \, dy = \frac{y^{n+1}}{n+1}$, we find that the integral of $y^a$ is $\frac{y^{a+1}}{a+1}$.

We evaluate this from $1$ to $3$. Applying the limits, we get:

$$\left[ \frac{y^{a+1}}{a+1} \right]_1^3 = \frac{3^{a+1}}{a+1} - \frac{1^{a+1}}{a+1}$$

Since $1$ raised to any power is $1$, this simplifies beautifully to:

$$\frac{3^{a+1} - 1}{a+1}$$

### The JEE Strategy: Tactical Thinking

We are given that the area is $\frac{364}{3}$. So, we set our expression equal to this value:

$$\frac{3^{a+1} - 1}{a+1} = \frac{364}{3}$$

Now, here is the secret to success in JEE Advanced: do not get bogged down in complex algebraic manipulation. The equation above is a classic example where testing the provided options is the superior strategy.

Let us test $a=5$. If $a=5$, the left-hand side becomes:

$$\frac{3^{5+1} - 1}{5+1} = \frac{3^6 - 1}{6}$$

Calculating $3^6$, we get $729$. So, we have:

$$\frac{729 - 1}{6} = \frac{728}{6}$$

Simplifying this fraction by dividing both numerator and denominator by $2$, we get $\frac{364}{3}$. It matches perfectly!

The logic holds, the math is sound, and we have arrived at the solution with precision and speed. The value of $a$ is **5**.

Animated Solution for Mathematics - Definite Integration: The odd natural number $a$ , such that the area of the region bounded by $y = 1, y = 3, x = 0, x = y^{a}$ is $\frac{364}{3}$ , equal to:

Select Answer:

Visualized Solution

Identifying the Boundaries

The Bounded Area

Setting Up the Integral

Substituting the Functions

Integrating the Function

Applying the Limits

Simplifying the Expression

Equating to Given Area

Strategy: Testing Options

Testing $a = 5$ (Part 1)

Testing $a = 5$ (Part 2)

Final Verification

The Sigma Insight: Area Bounded by Curves

Analyzing the Setup

The Power of Integration

The Integration Journey

The JEE Strategy

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Animated Solution for Mathematics - Definite Integration: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The odd natural number a, such that the area of the region bounded by y=1,y=3,x=0,x=ya is 3364​, equal to: