Question

The area (in sq.units) of the region A = {(x, y) ∈ R × R/0 ≤ x ≤ 3, 0 ≤ y ≤ 4, y ≤x² + 3x} is ______.

विकल्प

• `26/3`

• `59/6`

• `53/6`

• 8

Answer 1

The area (in sq.units) of the region A = {(x, y) ∈ R × R/0 ≤ x ≤ 3, 0 ≤ y ≤ 4, y ≤x² + 3x} is `underlinebb(59/6)`.

Explanation:

y = x² + 3x

= `(x + 3/2)^2 - 9/4`

Parabola with vertex `(-3/2, -9/4)` passing origin

`((-3)/2, (-9)/4){:(0 ≤ x ≤ 3),(0 ≤ y ≤ 4):}`

Required area OABC this can be divided into OCD + Rectangle DABC

Area OCD

For C point solve y = 4 and y = x² + 3x

x² + 3x = 4

x² + 3x – 4 = 0

(x + 4)(x – 1) = 0

x = –4, 1

`int_0^1(x^3 + 3x)dx + 4 xx 2 = [x^3/3 + (3x^2)/2]_0^1 + 8`

= `1/3 + 3/2 + 8`

= `(2 + 9 + 48)/6`

= `59/6` sq.units