Let f(x) be a non-negative continuous function such that the area bounded by the curve y = f(x), x-a

Q: Why can't I just integrate $f(x)$? We don't know what $f(x)$ is!

Exactly! That is the beauty of the problem. We are working backwards. We are given the *result* of the integration (the area), and we need to find the *integrand* (the function). We don't integrate; we differentiate the area function to recover the original function. It is like being given the distance traveled and needing to find the velocity.

Q: What happens to the $\pi/4$ in the lower limit when we differentiate?

Great observation. When we apply the Leibniz Rule, the derivative of the lower limit term involves the derivative of the constant $\pi/4$. Since the derivative of any constant is zero, that term vanishes. It is a classic JEE trap to make you overthink the lower limit, but remember: constants disappear under differentiation!

Q: Is the Leibniz Rule only for complex integrals?

Not at all! It is a fundamental tool for any integral where the limits depend on the variable of differentiation. It is the bridge between the area under a curve and the curve itself. Master this, and you master a huge chunk of calculus. It is not just a rule; it is a way to see the relationship between accumulation and instantaneous rate.

Question

Let $f(x)$ be a non-negative continuous function such that the area bounded by the curve $y = f(x)$, x-axis and the ordinates $x = \pi/4$ and $x = \beta > \pi/4$ is $[\beta \sin \beta + \frac{\pi}{4} \cos \beta + \sqrt{2} \beta]$. Then $f(\pi/2)$ is

Options:

(A) $(\frac{\pi}{4} + \sqrt{2} - 1)$

(B) $(\frac{\pi}{4} - \sqrt{2} + 1)$

(C) $(1 - \frac{\pi}{4} - \sqrt{2})$

(D) $(1 - \frac{\pi}{4} + \sqrt{2})$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing on the $x$-axis, looking at a curve $y = f(x)$. You have a vertical fence at $x = \pi/4$, and a movable fence at $x = \beta$.

The area trapped between these fences is not static; it grows as you slide the fence at $\beta$ to the right. We are given the 'result' of this growth—the area—and we must reverse-engineer the 'cause'—the function $f(x)$.

### The Bridge: Leibniz Rule

We start with the integral representation of the area $A(\beta)$:

$$A(\beta) = \int_{\pi/4}^{\beta} f(x) dx$$

We are told this area equals $\beta \sin \beta + \frac{\pi}{4} \cos \beta + \sqrt{2} \beta$. To find $f(\beta)$, we need to peel away the integral sign using the **Leibniz Rule** for differentiation under the integral sign.

This rule states that the derivative of an integral with respect to its upper limit is simply the integrand evaluated at that limit. It is the mathematical equivalent of reversing a process to see what created it.

### The Differentiation

Let us differentiate both sides with respect to $\beta$:

$$\frac{d}{d\beta} \left( \int_{\pi/4}^{\beta} f(x) dx \right) = \frac{d}{d\beta} \left( \beta \sin \beta + \frac{\pi}{4} \cos \beta + \sqrt{2} \beta \right)$$

On the left, the Leibniz Rule gives us $f(\beta)$. On the right, we differentiate each term individually. For the first term, $\beta \sin \beta$, we apply the **product rule**:

$$\frac{d}{d\beta}(\beta \sin \beta) = (\beta)(\cos \beta) + (\sin \beta)(1) = \sin \beta + \beta \cos \beta$$

The derivative of $\frac{\pi}{4} \cos \beta$ is $-\frac{\pi}{4} \sin \beta$, as $\frac{\pi}{4}$ is a constant coefficient. Finally, the derivative of $\sqrt{2} \beta$ is simply $\sqrt{2}$.

### The Final Reveal

Combining these pieces, we obtain the explicit function:

$$f(\beta) = \sin \beta + \beta \cos \beta - \frac{\pi}{4} \sin \beta + \sqrt{2}$$

Now, we evaluate this at $\beta = \pi/2$. Recalling the trigonometric identities $\sin(\pi/2) = 1$ and $\cos(\pi/2) = 0$, we substitute these values:

$$f(\pi/2) = \sin(\pi/2) + (\pi/2) \cos(\pi/2) - \frac{\pi}{4} \sin(\pi/2) + \sqrt{2}$$

$$f(\pi/2) = 1 + (\pi/2)(0) - \frac{\pi}{4}(1) + \sqrt{2}$$

The final result is:

**$$f(\pi/2) = 1 - \frac{\pi}{4} + \sqrt{2}$$**

Animated Solution for Mathematics - Definite Integration: Let $f (x)$ be a non-negative continuous function such that the area bounded by the curve $y = f (x)$ , x-axis and the ordinates $x = π /4$ and $x = β > π /4$ is $[β sin β + \frac{π}{4} cos β + 2 β]$ . Then $f (π /2)$ is

Select Answer:

Visualized Solution

Visualizing the Area $A (β)$

The Integral Representation

Equating with Given Expression

Applying Leibniz Rule

Differentiating the LHS

Differentiating RHS: Term 1

Differentiating RHS: Term 2 & 3

Combining to find $f (β)$

Substituting $β = π /2$

Evaluating Trigonometric Values

Final Calculation

The Sigma Insight: Area Bounded by Curves

Analyzing the Setup

The Bridge

The Differentiation

The Final Reveal

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