Use definite integrals to find the centroid of the region between the graphs of $f ( x ) = 3 x$ and the x-axis on the interval [0, 3].
A) (2, 4)
B) (1.5, 3)
C) (2, 3)
D) (0, 3)

Question

Quizplus · Accepted Answer

The centroid of the region is given by the formula:$$\overline{x} = \frac{1}{A} \int_a^b x f(x) dx$$$$\overline{y} = \frac{1}{A} \int_a^b \frac{1}{2} f(x)^2 dx$$where A is the area of the region.In this case, the area of the region is:$$A = \int_0^3 3x dx = \frac{9}{2}$$The x-coordinate of the centroid is:$$\overline{x} = \frac{1}{A} \int_0^3 x (3x) dx = \frac{1}{\frac{9}{2}} \int_0^3 3x^2 dx = \frac{2}{9} \left[ x^3 \right]_0^3 = 2$$The y-coordinate of the centroid is:$$\overline{y} = \frac{1}{A} \int_0^3 \frac{1}{2} (3x)^2 dx = \frac{1}{\frac{9}{2}} \int_0^3 \frac{9}{2} x^2 dx = \frac{2}{9} \left[ 3x^3 \right]_0^3 = 3$$Therefore, the centroid of the region is (2, 3).

Use Definite Integrals to Find the Centroid of the Region f(x)=3xf ( x ) = 3 xf(x)=3x

Use Definite Integrals to Find the Centroid of the Region $f ( x ) = 3 x$