Question

Area of the region bounded by the curve y = x², the X-axis and the lines x = 1 and x = 3 is ______.

विकल्प

• `3/26` sq. units

• 3 sq. units

• 26 sq. units

• `26/3` sq. units

Answer 1

Area of the region bounded by the curve y = x², the X-axis and the lines x = 1 and x = 3 is `underlinebb(26/3 sq. units)`.

Explanation:

1. Identify the Definite Integral

To find the area under a curve y = f(x) from x = a to x = b, we evaluate the definite integral of the function over that interval. In this case:

Function: y = x²

Lower Limit: x = 1

Upper Limit: x = 3

The integral is expressed as:

Area = `int_1^3 x^2 dx`

2. Calculate the Antiderivative

Applying the power rule for integration, `int x^n dx = x^(n + 1)/(n + 1)`, we find he antiderivative of x²:

`int x^2 dx = x^3/3`

3. Evaluate the Definite Integral

Substitute the upper and lower limits into the antiderivative and subtract:

Area = `[x^3/3]_1^3`

Area = `(3)^3/3 - (1)^3/3`

Area = `27/3 - 1/3`

Area = `26/3`