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=x2,x∈0,1, is rotated about the x-axis.
According to Table 5.5.1, the "formula" 2 π ∫abρ ⅆs becomes
2 π ∫01x21+ⅆⅆ x x22 ⅆx
=2 π ∫01x2 1+4 x2 ⅆx
=2 π 132⁢x⁢4⁢x2+1⁢8⁢x2+1−164arcsinh2 x01
An antiderivative for x21+4 x2 can be found, for example, by making the substitution x=tanθ/2. Methods such as this are developed in Chapter 6. The antiderivative is given in terms of the arcsinh function, which can also be expressed as a logarithm.
In Figure 5.5.1(a) the
tutor has been applied to the graph of y=x2 rotated about the x-axis. 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 returned, along with a floating-point equivalent. Note the SurfaceOfRevolution command at the bottom of the tutor. This command can return the graphs shown in Figures 5.5.1(a-b), the unevaluated integral, or the value of the surface area.
Figure 5.5.1(a) Surface of Revolution tutor
Student:-Calculus1:-SurfaceOfRevolution(x^2,x=0..1,partition=3,showsum=true, showsurface=false,output= plot,caption="",axes=frame,orientation=[-105,80,0],labels=[x,z,y],tickmarks=[2,,3],sumsurfaceoptions=[color=red]);
Figure 5.5.1(b) Segmentation into frustums
Figure 5.5.1(b) shows the surface of revolution segmented into three frustums, an image that can be generated in the tutor by selecting "Frustums" in the Display section, and then pressing the Display button. The default number of subintervals is 6, but this has been changed to 3 in the figure. In addition to the graph in Figure 5.5.1(b), the tutor also provides the following midpoint Riemann approximating sum.
π⁢∑i=1319⁢13⁢i-132+19⁢i2⁢4⁢i2-4⁢i+10 ≐ 3.885407601
Interactive evaluation of the surface-area integral
Expression palette: Definite-integral and derivative templates
Press the Enter key.
Context Panel: Evaluate and Display Inline
Context Panel: Approximate≻10 (digits)
2 π∫01x21+ⅆⅆ x x22 ⅆx
→at 10 digits
Application of the SurfaceOfRevolution command
Tools≻Load Package: Student Calculus 1
SurfaceOfRevolutionx2,x=0..1,output=integral = ∫012⁢π⁢x2⁢4⁢x2+1ⅆx
SurfaceOfRevolutionx2,x=0..1 = 132⁢π⁢−2⁢ln⁡2+18⁢5−ln⁡12+14⁢5
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