Question

The centroid of the plane region bounded by the graphs of \(y=f(x), y=0, x=0,\) and \(x=1\) is \(\left(\frac{5}{6}, \frac{5}{18}\right)\). Is it possible to find the centroid of each of the regions bounded by the graphs of the following sets of equations? If so, identify the centroid and explain your answer. (a) \(y=f(x)+2, y=2, x=0,\) and \(x=1\) (b) \(y=f(x-2), y=0, x=2,\) and \(x=3\) (c) \(y=-f(x), y=0, x=0,\) and \(x=1\) (d) \(y=f(x), y=0, x=-1,\) and \(x=1\)

Short answer

For part (a), the centroid is located at \(\left(\frac{5}{6}, 2+\frac{5}{18}\right)\). For part (b), the centroid is at \(\left(2 + \frac{5}{6},\frac{5}{18}\right)\). For part (c), the centroid is at \( \left( \frac{5}{6},-\frac{5}{18} \right) \). Finally, for part (d), the centroid is located at \(\left(0,\frac{5}{18}\right)\).

Step-by-step solution

Understand function transformation

Given that the centroid C for the plane region bounded by \(y=f(x), y=0, x=0,\) and \(x=1\) is at \(\left(\frac{5}{6}, \frac{5}{18}\right)\). When functions are transformed, these shifts affect the location of the centroid for the area bounded by the graphs of transformations.

Apply function transformation for part (a)

In this equation set \(y=f(x)+2, y=2, x=0, x=1\), the function \(f(x)\) is shifted two places upwards. This does not affect the x-coordinate of the centroid, but it does shift the y-coordinate two units upwards from the original centroid. So the new centroid is at \(\left(\frac{5}{6}, 2+\frac{5}{18}\right)\).

Apply function transformation for part (b)

In this equation set \(y=f(x-2), y=0, x=2, x=3\), the function \(f(x)\) is shifted two units to the right. This moves the x-coordinate of the centroid twos units to the right, but does not affect the y-coordinate. Therefore, the new centroid is \(\left(2 + \frac{5}{6},\frac{5}{18}\right)\).

Apply function transformation for part (c)

In this equation set \(y=-f(x), y=0, x=0, x=1\), the function is reflected about x-axis. This does not affect the x-coordinate of the centroid, but it reflects the y value with respect to x-axis. Hence, the new centroid is \( \left( \frac{5}{6},-\frac{5}{18} \right) \).

Apply function transformation for part (d)

In this equation set \(y=f(x), y=0, x=-1, x=1\), the region is expanded horizontally for a unit to each direction. This does not affect the y-coordinate of the centroid, but it makes x-coordinate to be at the middle of the region, hence the new centroid is \(\left(0,\frac{5}{18}\right)\).

Key concepts

Function Transformation

Understanding function transformation is crucial when dealing with the calculation of centroids in plane regions bounded by curves. A function transformation occurs when we apply certain operations to a function that alter its graph. These operations can include shifting, stretching, compressing, and reflecting.

For example, given the function transformation represented as \(y=f(x)+k\), the function will shift up by \(k\) units if \(k\) is positive or down if \(k\) is negative, without altering the shape of the graph. Similarly, a transformation like \(y=f(x-h)\) will shift the function to the right by \(h\) units if \(h\) is positive, or to the left if \(h\) is negative. Importantly, while these transformations change the position of the graph, they do not inherently change the 'spread' and 'balance' of the area under the curve, which are key factors in determining the location of the centroid.

Reflections are another type of transformation. Reflecting a function around the x-axis, denoted by \(y = -f(x)\), changes the sign of the y-values of all points on the graph. In all these cases, the x-coordinates of the centroids are influenced by horizontal shifts and reflections, whereas the y-coordinates are affected by vertical shifts and reflections.

When solving problems that involve the transformation of functions, one must thus consider how each type of transformation will change the location of the centroid in relation to the original function's graph.

Coordinates of Centroid

The coordinates of the centroid for a two-dimensional region are essentially the 'average' of the coordinates of all the points in that area. It's where the region would balance perfectly if it were a physical shape of consistent density. To find the centroid \((x_c, y_c)\) of a plane region, one typically integrates to find the first moments about the y-axis and x-axis and then divides by the total area of the region.

The centroid is vital in fields like mechanical engineering, statics, and material science, because it provides a single point that represents the average position of the entire area. If the area is uniform, each point is weighted equally, and thus, the centroid can be considered as the center of gravity. In mathematical terms, if \(A\) is the area of the region, the coordinates of the centroid are given by \(x_c = \frac{1}{A}\int x \, dA\) and \(y_c = \frac{1}{A}\int y \, dA\).

When the equations bounding the region are altered through function transformation, the coordinates of the centroid are recalculated accordingly. This involves considering the transformed equations and how they modify the bounds of integration, resulting in a new centroid's coordinates that reflect the updated shape and position of the region.

Graphical Representation of Functions

Graphical representation of functions plays an essential role in visualizing the effects of function transformations and understanding the location of centroids in plane regions. By graphing the original function and its transformations, one can clearly see how the region of interest changes with each modification. This visual aid is particularly helpful for comprehending complex changes like those involving reflections or simultaneous shifts.

The graphical method also allows students to verify their calculations by observing the symmetry and shape of the region. For instance, if a function \(f(x)\) is shifted vertically, one will see a corresponding upward or downward movement in the graph. This gives a visual confirmation that the y-coordinate of the centroid should also move in the same direction. Similarly, a horizontal shift will indicate a movement in the x-coordinate of the centroid.

When educating on this topic, it can be helpful to provide multiple examples of function graphs, before and after transformations, displaying how the centroid moves accordingly. The pairs of graphs visually represent the algebraic calculations and offer an intuitive understanding of the centroid's behavior in relation to shifts, stretches, and reflections. This visualization reinforces the conceptual knowledge allowing for a deeper understanding of the centroid's dependence on the geometry of the region.