Question

area of region bounded by [x]² = [y]² if x ∈ [1,5]​

Answer 1

Question:-

find area of region bounded by $\sf [x]^2 = [y]^2$ if $\sf x \in [1,5]$

Answer:-

(assuming that [x] and [y] denotes greatest integer function)

if $\sf 1 \leqslant x < 2 \implies [x] = 1 \implies [y] = \pm 1$

$\boxed{\sf \therefore y \in [-1,0) \cup [1,2)}$

if $\sf 2 \leqslant x < 3\implies{[x]} = 2 \implies [y] = \pm 2$

$\boxed{\sf \therefore y \in [-2,-1) \cup [2,3)}$

if $\sf 3 \leqslant x < 4 \implies [x] = 3 \implies [y] = \pm 3$

$\boxed{\sf \therefore y \in [-3,-2) \cup [3,4)}$

if $\sf 4\leqslant x < 5 \implies [x] = 4 \implies [y] = \pm 4$

$\boxed{\sf \therefore y \in [-4,-3) \cup [4,5)}$

now graph these all boxed values

( refer to Attachment)

from figure required area consist of 8 squares of of area unity i.e. 1 sq. unit.

so required area = 8 sq. units.

Additional Information:-

The area under a curve between two points can be found by doing a definite integral between the two points. To find the area under the curve y = f(x) between x = a and x = b, integrate y = f(x) between the limits of a and b. Areas under the x-axis will come out negative and areas above the x-axis will be positive.