Let R be the region bounded by the x-axis, the graph of y= √(x + 1), and the line x = 3. Find the area of the region R

Solution:

The region bounded by the  curve y= √(x+1) and the line x = 3 is shown in the diagram below. The shaded area lies above the x axis because the function y= √(x + 1) lies above the x - axis. Therefore the region R bounded by the curve and the line x = 3 is given by:

The Area ABC = \(\int_{-1}^{3}\sqrt{x+1}dx\)

= \([\frac{(x+1)^{\frac{1}{2}+1}}{\frac{3}{2}}]_{-1}^{3}\)

= \(\frac{2}{3}[(x+1)^{\frac{3}{2}}]_{-1}^{3}\)

= \(\frac{2}{3}[(3+1)^{\frac{3}{2}} - (-1 + 1)^{\frac{3}{2}}]\)

= \(\frac{2}{3}[(4)^{\frac{3}{2}} - (0)^{\frac{3}{2}}]\)

= \(\frac{2}{3}[\sqrt{64}]\)

= \(\frac{2}{3}[8]\)

= 16/3 units²

Let R be the region bounded by the x-axis, the graph of y= √(x+1), and the line x = 3. Find the area of the region R

Summary:

The area of the region R is 16/3 units².