Question

An ant is observed crawling on a sheet of paper along a straight line given by equation y = 2x − 4. Area of the surface covered by the ant bounded by y-axis, x-axis and x = 1 is ______.

पर्याय

• 1 sq. unit

• 3 sq. unit

• 2 sq. unit

• 4 sq. unit

Answer 1

An ant is observed crawling on a sheet of paper along a straight line given by equation y = 2x − 4. Area of the surface covered by the ant bounded by y-axis, x-axis and x = 1 is 3 sq. unit.

Explanation:

Let’s find the vertices of the region:

Intersection with y-axis (x = 0):

y = 2x − 4

= 2(0) − 4

= −4

Point is (0, −4)

At x = 1:

y = 2(1) − 4

= −2

The boundaries: x = 0, x = 1, the x-axis (y = 0), and the line y = 2x − 4.

Area is given by the definite integral of the function. Since the region is entirely below the x-axis, we take the absolute value (or negate the integral):

Area = `|int_0^1 (2x - 4) dx|`

Perform the integration:

`int(2x - 4) dx = [x^2 - 4x]`

Apply limits from 0 to 1:

[(1)² − 4(1)] − [(0)² − 4(0)]

= (1 − 4) − (0)

= −3

Since area cannot be negative:

Area = |−3| = 3 sq. units