Example 5: Rotating a Curve About X-axis and Y-axis

Let C be the curve

y = x⁴, 0 < x < 1.

(see Figure 6.4.15)

Figure 6.4.15               

Set up an integral for the surface area generated by rotating the curve C about (a) the y-axis, (b) the x-axis (see Figure 6.4.16).

(a)

Figure 6.4.16 (a)

dy/dx = 4x³

A =

We cannot evaluate this integral, so we leave it in the above form. The Trapezoidal Rule can be used to get approximate values. When Δx = 1/10 the Trapezoidal Approximation is

A ~ 6.42, error ≤ 0.26.

(b)

Figure 6.4.16(b)      

A =

The Trapezoidal Approximation when Δx = 1/10 is A ~ 3.582 error ≤ 0.9.