Chapter 5: Applications of Integration
Section 5.5: Surface Area of a Surface of Revolution
Calculate the surface area of the surface of revolution formed when the graph of y=e−x,x∈0,2, is rotated about the y-axis.
According to Table 5.5.1, the "formula" 2 π ∫abρ ⅆs becomes
2 π ∫02x 1+ⅆⅆ x e−x2 ⅆx = 2 π ∫02x 1+e−2 x ⅆx ≐ 13.23912130
Maple cannot provide an antiderivative for the integrand x 1+e−2 x, so the definite integral is evaluated numerically. While a Riemann sum with approximating rectangles could be used, more efficient methods will be seen in Chapter 6.
In Figure 5.5.5(a) the
tutor has been applied to the graph of y=e−x rotated about the y-axis. Note how e−x is entered into the tutor as exp−x. Moreover, to calculate with the exponential function, the exponential e must be entered from a palette (e.g., Common Symbols) or via command completion (Tools menu, or the Escape key).
The Plot Options section has been used to impose one-to-one scaling, and the frame axis style.
An exact representation of the value of the surface-area integral is not available, so the tutor provided just a floating-point approximation.
Figure 5.5.5(a) Surface of Revolution tutor
Interactive evaluation of the surface-area integral
Expression palette: Definite-integral and derivative templates
Press the Enter key.
Context Panel: Approximate≻10 (digits)
2 π∫02x 1+ⅆⅆ x ⅇ−x2 ⅆx
→at 10 digits
Application of the SurfaceOfRevolution command
Tools≻Load Package: Student Calculus 1
SurfaceOfRevolutionⅇ−x,x=0..2,axis=vertical = ∫022⁢π⁢x⁢1+ⅇ−x2ⅆx→at 10 digits13.23912130
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