NIMCET 2010 — Mathematics PYQ

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Question

NIMCET 2010 — Mathematics PYQ

NIMCET | Mathematics | 2010

The area between the curves $y = 2 - x^2$ and $y = x^2$ is:

Choose the correct answer:

Explanation

Solution

1. Find the Points of Intersection

To find where the two curves meet, we set the equations equal to each other:

2 - x^2 = x^2

2 = 2x^2

x^2 = 1

x = \pm 1

The curves intersect at $x = -1$ and $x = 1$ .

2. Set up the Definite Integral

The area $A$ between two curves $y_{upper}$ and $y_{lower}$ from $x = a$ to $x = b$ is given by:

A = \int_{a}^{b} (y_{upper} - y_{lower}) \, dx

In the interval $[-1, 1]$ , the curve $y = 2 - x^2$ is above $y = x^2$ .

A = \int_{-1}^{1} ((2 - x^2) - x^2) \, dx

A = \int_{-1}^{1} (2 - 2x^2) \, dx

3. Evaluate the Integral

Since the function is even, we can simplify by integrating from $0$ to $1$ and doubling the result:

A = 2 \int_{0}^{1} (2 - 2x^2) \, dx

A = 2 \left[ 2x - \frac{2x^3}{3} \right]_{0}^{1}

A = 2 \left[ \left( 2(1) - \frac{2(1)^3}{3} \right) - (0) \right]

A = 2 \left[ 2 - \frac{2}{3} \right]

A = 2 \left[ \frac{6 - 2}{3} \right]

A = 2 \left[ \frac{4}{3} \right]

A = \frac{8}{3}

Correct Option:

The correct option is (a) $8/3$ .