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NDA 2024
Exam
NDA
Year
2024
Subject
Mathematics
Chapter
Definite Integration

NDA 2024 · Mathematics · Definite Integration

Paragraph: Let f(x)=|x^2-x-2| Question: What is ∫_0^2 f(x) d x...

Paragraph: Let f(x)=∣x2−x−2∣f(x)=\left|x^2-x-2\right|
Question: What is ∫02f(x)dx\int_0^2 f(x) d x equal to?

Options

  1. A0
  2. B1
  3. C53\frac{5}{3}
  4. D103\frac{10}{3}

Correct answer

D. 103\frac{10}{3}

Step-by-step solution

 Given f(x)=∣x2−x−2∣{x2−x−2;x∈(−∞,−1]∪[2,∞)−(x2−x−2);x∈[−1,2]\begin{aligned} & \text{ Given } f(x)=\left|x^2-x-2\right| \\ & \left\{\begin{array}{l}x^2-x-2 ; x \in(-\infty,-1] \cup[2, \infty) \\ -\left(x^2-x-2\right) ; x \in[-1,2]\end{array}\right.\end{aligned} Let I=∫02f(x)=−∫02(x2−x−2)dx=−[x33−x22−2x]02=−(83−42−4)+0=−(8−183)=103\begin{aligned} I & =\int_0^2 f(x)=-\int_0^2\left(x^2-x-2\right) d x=-\left[\frac{x^3}{3}-\frac{x^2}{2}-2 x\right]_0^2 \\ & =-\left(\frac{8}{3}-\frac{4}{2}-4\right)+0 \\ & =-\left(\frac{8-18}{3}\right)=\frac{10}{3}\end{aligned}
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