91Ó°ÊÓ

To find the centroid of the semicircular arc, we can use Pappus' theorem, specifically the first Pappus centroid theorem relating the area of a curve with polar coordinates, its arc length, and centroid. According to Pappus' theorem, the area of the curve can be expressed as \(A = 2 \pi X \cdot L\), where \(X\) is the x-coordinate of the centroid and \(L\) is the arc length of the curve. However, since we only have a semicircular arc, we need to make some adjustments. The length of the semicircular arc is given by \(L = \pi a\) and the total area of the semicircle is \(A = \frac{1}{2} \pi a^2\). In the case of a full circle, the centroid would be at the origin. However, for a semicircular arc, the centroid's x-coordinate would still be zero. Therefore, we just need to find the y-coordinate of the centroid. Let's call the y-coordinate of the centroid \(Y\), then by Pappus' theorem, we have: \[\frac{1}{2} \pi a^2 = 2 \pi (0) Y L \] \[A = 2 \pi Y L\] Now, let's solve for \(Y\).

Using our expressions for the area and arc length, and the Pappus' theorem equation we derived, we can solve for \(Y\). \[ \frac{1}{2} \pi a^2 = 2 \pi Y (\pi a)\] Now, we solve for Y: \[Y = \frac{\frac{1}{2} \pi a^2}{2 \pi^2 a} = \frac{a}{4 \pi}\] So, the centroid of the semicircular arc is at coordinates \((0, \frac{a}{4 \pi})\).

Now that we have the centroid of the arc, we need to find the surface area generated by revolving this arc about the line given by \(y = a\). We can again use Pappus' theorem, specifically the second Pappus centroid theorem relating the volume of a solid of revolution with the curve's centroid. The second Pappus centroid theorem states that the surface area of a solid generated by revolving a curve about a line is \( S = 2 \pi \cdot Y \cdot L\), where \(S\) is the surface area of the solid, \(Y\) is the distance from the centroid to the line of revolution, and \(L\) is the arc length of the curve. First, we need to find the distance between the centroid and the line \(y = a\). The y-coordinate of the centroid is \(\frac{a}{4 \pi}\), so the distance between the centroid and the line of revolution is the difference in y-values: \[Y = a - \frac{a}{4 \pi}\] Now, plug this value for \(Y\) into the Pappus' theorem formula to find the surface area:

Using our formula for the surface area from Pappus' theorem, we can plug in our values for \(Y\) and \(L\) to compute the surface area: \[ S = 2 \pi \left( a - \frac{a}{4 \pi} \right) (\pi a) \] Simplifying this expression: \[ S = 2 \pi^2 a^2 - \frac{1}{2} \pi^2 a^2 \] \[ S = \frac{1}{2} \pi^2 a^2 \] The surface area generated by revolving the semicircular arc about the line \(y = a\) is \(\frac{1}{2} \pi^2 a^2\).