If y=f(x) makes +ve intercept of 2 and 0 unit on x and y axes and encloses an area of 3/4 square uni

Q: Wait, why is $f(0)=0$? The question says y-intercept is 0.

The y-intercept is the point where the curve crosses the y-axis, which occurs at $x=0$. If the y-intercept is 0, the point is $(0,0)$, so $f(0)=0$. It is a simple but crucial observation!

Q: Why can't we just integrate $f'(x)$ directly?

Because we do not know the explicit form of $f(x)$! We only know its integral (the area). Integration by parts is powerful because it transforms the unknown $\int x f'(x) dx$ into the known $\int f(x) dx$.

Q: Why does the boundary term $[xf(x)]_0^2$ vanish?

Evaluate it at the limits: $2f(2) - 0f(0)$. Since the curve hits the x-axis at $x=2$, $f(2)=0$. And since it passes through the origin, $f(0)=0$. Both terms become zero, leaving only the integral part.

Question

If $y=f(x)$ makes +ve intercept of 2 and 0 unit on x and y axes and encloses an area of 3/4 square unit with the axes then $\int_{0}^{2} xf'(x) dx$ is

Options:

(A) 3/2

(B) 1

(C) 5/4

(D) -3/4

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing on the Cartesian plane. You see a curve, $y=f(x)$, dancing between the axes. The problem provides two vital clues: it hits the x-axis at $x=2$ and the y-axis at $y=0$.

In the language of functions, these are not just points; they are boundary conditions. The x-intercept at $2$ tells us that $f(2)=0$, and the y-intercept at $0$ tells us that $f(0)=0$.

These two points are the anchors of our journey. We do not know the shape of the curve, but we know exactly where it starts and where it ends.

### The Area Constraint

Now, look at the space trapped between the curve and the x-axis. The problem states this area is $\frac{3}{4}$ square units.

Mathematically, this is expressed as the definite integral:

$$ \int_{0}^{2} f(x) dx = \frac{3}{4} $$

This is our treasure map. We do not need to know the function $f(x)$ itself; we only need to know how it behaves under the integral sign.

### The Integration by Parts Strategy

We are asked to evaluate the integral:

$$ I = \int_{0}^{2} x f'(x) dx $$

Whenever you see a product of a polynomial and a derivative inside an integral, your intuition should immediately suggest **Integration by Parts**. The formula is $\int u dv = uv - \int v du$.

We choose $u=x$ and $dv=f'(x)dx$. This is a strategic choice because differentiating $u=x$ yields $du=dx$, which simplifies the integral, while integrating $dv=f'(x)dx$ yields $v=f(x)$, returning us to the function we know.

### The Vanishing Boundary

Let us apply the formula:

$$ I = [x f(x)]_{0}^{2} - \int_{0}^{2} f(x) dx $$

Now, consider the first term, $[x f(x)]_{0}^{2}$. This evaluates to $2f(2) - 0f(0)$.

Recall our anchors from the beginning: $f(2)=0$ and $f(0)=0$. The entire boundary term becomes $2(0) - 0(0) = 0$. It vanishes!

### The Final Reveal

We are left with the simplified expression:

$$ I = 0 - \int_{0}^{2} f(x) dx $$

Since we already know that $\int_{0}^{2} f(x) dx = \frac{3}{4}$, we substitute this value into our equation.

Therefore, the final result is:

$$ I = -\frac{3}{4} $$

This is the essence of advanced mathematics: identifying the right tool, respecting the boundary conditions, and trusting the process.

Animated Solution for Mathematics - Definite Integration: If $y = f (x)$ makes +ve intercept of 2 and 0 unit on x and y axes and encloses an area of 3/4 square unit with the axes then $\int_{0}^{2} x f^{'} (x) d x$ is

Select Answer:

Visualized Solution

Visual Anchor: Identifying Intercepts

The Area Constraint

Defining the Goal

Logic Bridge: Integration by Parts

Raw Setup: Substitution

Atomic Compute: Boundary Evaluation

Atomic Compute: Final Calculation

The Way Forward

The Sigma Insight: Fundamental Theorem & Properties of Definite Integrals

Analyzing the Setup

The Area Constraint

The Integration by Parts Strategy

The Vanishing Boundary

The Final Reveal

Similar Questions

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Select Answer:

Visualized Solution

Visual Anchor: Identifying Intercepts

The Area Constraint

Defining the Goal

Logic Bridge: Integration by Parts

Raw Setup: Substitution

Atomic Compute: Boundary Evaluation

Atomic Compute: Final Calculation

The Way Forward

The Sigma Insight: Fundamental Theorem & Properties of Definite Integrals

Analyzing the Setup

The Area Constraint

The Integration by Parts Strategy

The Vanishing Boundary

The Final Reveal

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