What is ₀²f(x)dx equal to

Correct Answer is : 10/310/310/3

Correct Answer is : 10/310/310/3

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UPSC NDA II 2024 Mathematics Paper

Question Numbers: 91-92

Direction : Consider the following for the items that follow :

Let

f(x) = \left| x^{2} - x - 2 \right|

Question : 91 of 120

Marks: +1, -0

What is

\rm \int_0^2f(x)dx

equal to

Solution:

Concept:

The integral of an absolute value requires splitting the interval where the expression inside is positive or negative.

Explanation:

First, factor the quadratic:

x^2 - x - 2 = (x-2)(x+1)

[0,2]

, the expression is

\le 0

(zero at

x=2

, negative elsewhere).

|x^2 - x - 2| = -(x^2 - x - 2)

[0,2]

Thus the integral becomes:

\displaystyle I = \int_0^2 f(x)\,dx = -\int_0^2 (x^2 - x - 2)\,dx

Evaluate the antiderivative:

-\left[ \frac{x^3}{3} - \frac{x^2}{2} - 2x \right]_0^2

Compute at

x=2

\frac{8}{3} - \frac{4}{2} - 4 = \frac{8}{3} - 2 - 4 = \frac{8}{3} - 6 = \frac{8}{3} - \frac{18}{3} = -\frac{10}{3}

x=0

it is

0

I = -\left( -\frac{10}{3} - 0 \right) = \frac{10}{3}

Answer:

\displaystyle \frac{10}{3}

(Option D).

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