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JEE Mains · Maths · STD 12 - 8. Application and integration
Let \(f:\left[ { - 2,3} \right] \to \left[ {0,\infty } \right)\) be a continuous function such that \(f(1-x) = f(x)\) for all \(x \in \left[ { - 2,3} \right]\) . If \(R_1\) is the numerical value of the area of the region bounded by \(y =f (x), x = -2, x = 3\) and the axis of \(x\) and \({R_2} = \int\limits_{ - 2}^3 {x\,f\left( x \right)} dx\) , then
- A \(3R _1= 2R_2\)
- B \(2R _1= 3R_2\)
- C \(R _1= R_2\)
- D \(R _1= 2R_2\)
Answer & Solution
Correct Answer
(D) \(R _1= 2R_2\)
Step-by-step Solution
Detailed explanation
We have \({{\rm{R}}_2} = \int\limits_{ - 2}^3 {xf(x)dx} \) \( = \int\limits_{ - 2}^3 {(1 - x)f(1 - x)dx} \) \(\left[ {{\rm{Using}}\int\limits_a^b {f\left( x \right)dx} = \int\limits_a^b {f\left( {a + b - x} \right)dx} } \right]\)…
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