Area Under The Curve-JEE-Mains-PYQ’s

Mathematics Questions

If the area of the region $\{(x, y): 1 + x^2 \leq y \leq \min \{x + 7, 11 - 3x\}\}$ is $A$ , then $3A$ is equal to:

(A) 50

(B) 46

(D) 47

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If the area of the region bounded by the curves $y = 4 - \frac{x^2}{4}$ and $y = \frac{x - 4}{2}$ is equal to $\alpha$ , then $6\alpha$ equals

(A) 210

(B) 250

(D) 220

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Let $f: [0, \infty) \to \mathbb{R}$ be a differentiable function such that

$f(x) = 1 - 2x + \int_0^x e^{x - t} f(t) \, dt \quad \text{for all } x \in [0, \infty).$

Then the area of the region bounded by $y = f(x)$ and the coordinate axes is

(A) $\sqrt{5}$

(B) 2

(D) $\frac{1}{2}$

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The area of the region $\{(x, y): |x - y| \leq y \leq 4\sqrt{x}\}$ is

(A) $\frac{512}{3}$

(B) $\frac{2048}{3}$

(D) $\frac{1024}{3}$

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Let the area enclosed between the curves $|y| = 1 - x^2$ and $x^2 + y^2 = 1$ be $\alpha$ . If $9\alpha = \beta\pi + \gamma$ , $\beta, \gamma$ are integers, then the value of $|\beta - \gamma|$ equals: