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STD 12 - 8. Application and integration — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 12 - 8. Application and integration4 Marks
MCQ
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
  • A
    $R_1=2 R_2$
  • B
    $R_1=3 R_2$
  • ✓
    $2 R_1=R_2$
  • D
    $3 R_1=R_2$
✓

Answer

Correct option: C.
$2 R_1=R_2$
c
$R_1=\int_{-1}^2 x f(x) d x=\int_{-1}^2(2-1-x) f(2-1-x) d x$

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

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

Hence $2 R_1=\int_{-1}^2 f(x) d x=R_2$.

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