Sketch the graph of y = √ (x + 1) in [0, 4] and determine the area of the region enclosed by the curve, the x - axis and the lines x = 0, x = 4

Question

Sketch the graph of y = \(\sqrt{x+1}\) in [0, 4] and determine the area of the region enclosed by the curve, the x - axis and the lines x = 0, x = 4

Answer 1

Given equations are:

x – axis ...... (1)

x = 0 ...... (2)

x = 4 ...... (3)

And y = \(\sqrt{x+1},\) 0 ≤ x ≤ 4 ...... (4)

equation (4) represents a half parabola with vertex at ( – 1, 0) and passing through (4, 0) on x – axis, equation (3) represents a line parallel to y - axis at a distance of 4 units and equation (2) represents y - axis.

A rough sketch is given as below: -

We have to find the area of the shaded region.

Required area

= shaded region AOBC

(the area can be found by taking a small slice in each region of width Δx, then the area of that sliced part will be yΔx as it is a rectangle and then integrating it to get the area of the whole region)

\(=\int^4_0 y\,dx\) (As x is between (0, 4) and the value of y varies)

Substitute u = x + 1 \(\Rightarrow\) dx = du

So the above equation becomes,

Hence the area of the region enclosed by the curve, the x - axis and the lines x = 0, x = 4 is equal to \(\frac{2(\sqrt{5^2}-1)}{3}\) square units.