This is an excellent question and certainly requires some tedious calculations. Let me try to break it down clearly.
First, note that volume is approximately the surface area multiplied by the ``thickness'' of the solid; in the notation of this problem, we have (Volume) ≈ (Surface Area) * (0.015 inches) * (conversion factor) --- the conversion factor is present since x and y are in feet while the thickness is in inches. This formula explains why we're interested in finding the surface area of the light bulb.
Next, we move on to determining the surface area; after we know the surface area, we'll return to the formula above to plug in the value we calculated. The general formula for surface area of a curve rotated about the x-axis is
(Surface Area) = (integral from a to b) (2πy) (arc length) dx
In this formula, the bounds 'a' and 'b' are 0 and 1/3 as expressed in the problem (we can also check that this is where the curve intersects the x-axis, though this is not necessary to know for this problem). The 'y' in the '2πy' factor is the radius. In particular, the 2πy term is actually the circumference of the circle when we rotate the point (x,y) about the x-axis. From the problem, we know the light bulb has equation y=(1/3)x1/2-x3/2 which means this '2πy' factor is (2π ( (1/3)x1/2-x3/2 ) ). Our next step is to determine the arc length formula so we can evaluate the integral.
Recall the formula for arc length is given by √( 1+(f'(x))2 ). Since y=(1/3)x1/2-x3/2, we can check that y'=(1/6)x-1/2-(3/2)x1/2. Here, it gets a little tedious, but if we calculate (y')2, we find that (y')2=(1/36)x-1-(1/2)+(9/4)x1. The important part to notice about the square of the derivative is that when we add 1 to it, the negative in front of the (1/2) becomes a positive. That is, we have 1+(y')^2=(1/36)x-1+(1/2)+(9/4)x1. The similarity between the two expressions is not a coincidence: this last term factors as ( (1/6)x-1/2+(3/2)x1/2 )2 --- note the similarity of the two expressions (y')2 and 1+(y')2. Importantly, observe that 1+(y')2 is an entire expression squared. When we take the square root of a squared expression, the two operations cancel out. It follows that (arc length) = (1/6)x-1/2+(3/2)x1/2.
Bringing everything together, we have
(Surface Area) = (integral from 0 to 1/3) (2π ( (1/3)x1/2-x3/2 ) ) ( (1/6)x-1/2+(3/2)x1/2 ) dx
The 2π is a constant that can be pulled out of the integral and we're left with a product of two binomials. Using the FOIL method for multiplying the binomials and combining like terms, we have
2π(integral from 0 to 1/3) (1/18) + (1/3) x - (3/2) x2 dx
Using the power rule for integration to evaluate each term and simplifying, this works out nicely to π/27. It's worth noting here that this quantity is in square feet. Converting this to square inches is a matter of multiplying by (144 in2)/(1 ft2), yielding 16π/3.
Thus, we have finally found the surface area. Returning to the original formula, the volume is approximately (16π/3)*(0.015) in3, which numerically is about 0.2513 cubic inches.
