Question

Sketch the region bounded by the curves y=3x^3, x+y=4, and y=0. Find the coordinates of the centroid.

Answer 1

You can start by sketching the region in the xy plane.
y = 3x^3 is a cubic function.
x + y = 4 is a line, you can solve for y to find slope/y-intercept on the plot
y = 0 is a horizontal line, but it is more formally known as the x-axis.

Answer 2

With the sketch, you can determine:
points of intersection
domain of integration
a rough idea of where the centroid might be (for checking your answer validity)

From there, you would need to find the area of the region. This can be done by integrating over the region's x length with the integrand being the difference of the top curve on the region and the bottom. It is important to split the domain of integration where the two functions, x+y = 4 and y = 3x^3 may cross because only one is the boundary for the region!

Answer 3

To find the coordinates of the centroid, then, we have two formulas to use -- one for each coordinate.

\( \displaystyle \bar{x} = \frac{1}{A} \int_{a}^{b} x \left( f(x) - g(x) \right) \; \text{d}x \)
\( \displaystyle \bar{y} = \frac{1}{A} \int_{a}^{b} \frac{1}{2} \left( f(x) + g(x) \right) \left( f(x) - g(x) \right)\; \text{d} x \)
Or simplify y bar as f^2 - g^2 , that may make it easier.

Answer 4

Does it matter which of x+y = 4 and y = 3x^3 I set as f(x) and g(x)?

Answer 5

Apologies for late response;

You should always split the domain so that f(x) will always be the smaller of the two functions (that is, the one that is actually used on the boundary). So, say [a, c] u [c, b], on one side f(x) could be the line equation. The other side, we would have the cubic equation. In the grand scheme, we would integrate over [a,c] x(f(x) - g(x)) dx and add it to the integral over [c,b] with the correct f(x) in the same spot.