Question

[Technology Exercise] 75. Find, to two decimal places, the x-coordinate of the centroid of the region in the first quadrant bounded by the x-axis, the curve y = arctan(x), and the line x = √3.

Accepted Answer

Identify the region bounded by the x-axis (y = 0), the curve y = arctan(x), and the vertical line x = \sqrt{3} in the first quadrant. This region lies between x = 0 and x = \sqrt{3}. Recall that the x-coordinate of the centroid $$ \bar{x}  of a $$region bounded by curves can be found using the formula:

$$
\bar{x} = \frac{1}{A} \int_a^b x \cdot f(x) \, dx
$$

where $$ A = \int_a^b f(x) \, dx  is $$the area under the curve from $$ a  to$$ $$ b $$, and $$ f(x)  is $$the function defining the upper boundary of the region. Calculate the area $$ A  of $$the region by integrating the function $$ f(x) = \arctan(x) $$ from 0 to $$ \sqrt{3} $$:

$$
A = \int_0^{\sqrt{3}} \arctan(x) \, dx
$$ Set up the integral for the moment about the y-axis (needed for $$ \bar{x} $$):

$$
M_y = \int_0^{\sqrt{3}} x \cdot \arctan(x) \, dx
$$ Finally, express the x-coordinate of the centroid as:

$$
\bar{x} = \frac{M_y}{A} = \frac{\int_0^{\sqrt{3}} x \cdot \arctan(x) \, dx}{\int_0^{\sqrt{3}} \arctan(x) \, dx}
$$

Evaluate these integrals (using integration by parts if necessary) and then divide to find $$ \bar{x} .$$

[Technology Exercise] 75. Find, to two decimal places, the x-coordinate of the centroid of the region in the first quadrant bounded by the x-axis, the curve y = arctan(x), and the line x = √3.

Verified video answer for a similar problem:

Key Concepts

Centroid of a Region

Definite Integrals for Area and Moments

Inverse Trigonometric Functions - arctan(x)