COMPASSLearning

Talk to us Register Login

JEE Main · 2024 · 29 Jan 2024 · Shift 2 · Area Under Curves

Let the area of the region

{(x, y) : 0 \leq x \leq 3, 0 \leq y \leq min {x^{2} + 2, 2 x + 2}}

be A. Then 12A is equal to ______.

Answer: 164

Solution

Compare

x^{2} + 2

and

2 x + 2

:

x^{2} + 2 - (2 x + 2) = x^{2} - 2 x = x (x - 2)

, negative for

0 < x < 2

and positive for

2 < x < 3

.

So

min {x^{2} + 2, 2 x + 2} = x^{2} + 2

on

[0, 2]

and

= 2 x + 2

on

[2, 3]

.

A = \int_{0}^{2} (x^{2} + 2) d x + \int_{2}^{3} (2 x + 2) d x

= [\frac{x ^{3}}{3} + 2 x]_{0}^{2} + [x^{2} + 2 x]_{2}^{3} = (\frac{8}{3} + 4) + [(9 + 6) - (4 + 4)]

= \frac{8}{3} + 4 + 7 = \frac{8}{3} + 11 = \frac{41}{3}

.

12 A = 12 \cdot \frac{41}{3} = 164

.

JEE Main · 2024 · 29 Jan 2024 · Shift 2 · Integer answer

Practise more: all JEE Main PYQs on Area Under Curves →

More JEE Main questions on Area Under Curves

The area of the region, inside the ellipse $x^{2} + 4 y^{2} = 4$ and outside the region bounded by the curves $y = ∣ x ∣ - 1$ and $y = 1 - ∣ x ∣$ , is :
JEE Main 2026View solution →
If the area of the region ${(x, y) : 1 - 2 x \leq y \leq 4 - x^{2}, x \geq 0, y \geq 0}$ is $\frac{α}{β}$ , $α, β \in N$ , $g cd (α, β) = 1$ , then the value of $(α + β)$ is :
JEE Main 2026View solution →
Let the line $x = - 1$ divide the area of the region ${(x, y) : 1 + x^{2} \leq y \leq 3 - x}$ in the ratio $m : n, g cd (m, n) = 1$ . Then $m + n$ is equal to
JEE Main 2026View solution →
The area of the region
$A = {(x, y) : 4 x^{2} + y^{2} \leq 8 and y^{2} \leq 4 x}$ is :
JEE Main 2026View solution →
Let the area of the region bounded by the curve $y = max {sin x, cos x}$ , lines $x = 0, x = \frac{3 π}{2}$ , and the x-axis be A. Then, $A + A^{2}$ is equal to ______.
JEE Main 2026View solution →
The area of the region enclosed between the circles $x^{2} + y^{2} = 4$ and $x^{2} + (y - 2)^{2} = 4$ is :
JEE Main 2026View solution →