Let f:[-1,2] arrow[0, ∞) be a continuous function such that f(x)=f(1-x) for all x ∈[-1,2] . Let R_1=∫_ -1 ^2 x f(x) d x and R_2 be the area of the region bounded by y=f(x), x=-1, x=2 and the X -axis. Then,

Correct answer: C. 2 R_1=R_2 Here, R_1=∫_ -1 ^2 x f(x) d x Using, ∫_a^b f(x) d x=∫_a^b f(a+b-x) d x aligned & R_1=∫_ -1 ^2(1-x) f(1-x) d x, & .R_1=∫_ -1 ^2(1-x) f(x) d x (1-x)] aligned Given, R_2 is area bounded by aligned & f(x), x-1 and x=2 & R_2=∫_ -1 ^2 f(x) d x aligned Adding Eqs. (i) and (ii), we get 2 R_1=∫_ -1 ^2 f(x) d x From Eqs. (iii) and (iv), we get 2 R_1=R_2

JEE Advanced 2011Paper 2

Exam: JEE Advanced
Year: 2011
Paper: Paper 2
Subject: Mathematics
Chapter: Definite Integration

JEE Advanced 2011 · Paper 2 · Mathematics · Definite Integration

Let f:[-1,2] arrow[0, ∞) be a continuous function such that...

Let

f:[-1,2] \rightarrow[0, \infty)

be a continuous function such that

f(x)=f(1-x)

for all

x \in[-1,2]

. Let

R_1=\int_{-1}^2 x f(x) d x

and

R_2

be the area of the region bounded by

y=f(x), x=-1, x=2

and the

X

-axis. Then,

Options

A $R_1=2 R_2$
B $R_1=3 R_2$
C $2 R_1=R_2$
D $3 R_1=R_2$

Correct answer

C. $2 R_1=R_2$

Step-by-step solution

Here,

R_1=\int_{-1}^2 x f(x) d x

Using,

\quad \int_a^b f(x) d x=\int_a^b f(a+b-x) d x

\begin{aligned} & R_1=\int_{-1}^2(1-x) f(1-x) d x, \\ \therefore \quad & \left.R_1=\int_{-1}^2(1-x) f(x) d x \quad \ldots(1-x)\right] \end{aligned}

Given,

R_2

is area bounded by

\begin{aligned} & f(x), x-1 \text{\text{ and } } x=2 \\ \therefore & R_2=\int_{-1}^2 f(x) d x \end{aligned}

Adding Eqs. (i) and (ii), we get 2 R_1=\int_{-1}^2 f(x) d x

\therefore

From Eqs. (iii) and (iv), we get 2 R_1=R_2

Practice Definite Integration on Quantrex Academy →

More from Definite Integration

The value of the definite integral ∫_ 0 ^ 2 1 3^ (x+3)/(2) \, dx is JEE Advanced 2026 The value of the integral ∫_ 0 ^ 2 x(x^2+x+1) ( x+1 )( x^4+x^2+1 ) \, dx is equal to: JEE Main 2026 If ∫_ π/6 ^ π/4 ( (x-(π)/(3)) (x+(π)/(3))+1)dx = α _e( 3 -1) , then 9α^2 is equal to… JEE Main 2026 Let A = bmatrix 1 & 3 & -1 2 & 1 & α 0 & 1 & -1 bmatrix be a singular matrix. Let f(x) =… JEE Main 2026 Let f: ℝ arrow ℝ be such that f(xy) = f(x)f(y) , for all x, y ∈ ℝ and f(0) ≠ 0 . Let g… JEE Main 2026 The value of the integral ∫ _ -1 ^ 1 ((x^3 + |x| + 1)/(x^2 + 2|x| + 1)) dx is equal to : JEE Main 2026 The value of the integral ∫_ -π/4 ^ π/4 ( 32 ^4 x 1 + e^ x )dx is: JEE Main 2026 Let (2^ 1-a + 2^ 1+a ) , f(a) , (3^a + 3^ -a ) be in A.P. and α be the minimum value of… JEE Main 2026 Let f: [1, ∞) arrow R be a differentiable function defined as f(x) = ∫_1^x f(t)\,dt +… JEE Main 2026 The value of the integral ∫_0^∞ ( _e(x))/(x^2 + 4)\,dx is: JEE Main 2026 The value of the integral ∫_ π/6 ^ π/3 ((4 - ^2 x)/( ^4 x)) dx is: JEE Main 2026 The integral ∫_ 0 ^ 1 ^ -1 (1+x+x^2)dx is equal to: JEE Main 2026 Let f be a twice differentiable function such that f(x)=∫_ 0 ^ x (t-x)dt-∫_ 0 ^ x f(t)… JEE Main 2026 Let ∫_ -2 ^ 2 (| x| + [x x])\,dx = 2(3 - 2) + β , where [·] is the greatest integer… JEE Main 2026 The value of ∫_0^ 20π ( ^4 x + ^4 x) \, dx is equal to: JEE Main 2026 Let [·] denote the greatest integer function. Then the value of ∫_0^3 ( e^x + e^ -x [x]!… JEE Main 2026 Full Definite Integration list Definite Integration JEE Maths PYQs JEE Mathematics PYQs All JEE Advanced PYQs