Chapter 5: Applications of Integration
Section 5.5: Surface Area of a Surface of Revolution
Example 5.5.2
Calculate the surface area of the surface of revolution formed when the graph of y=x2,x∈0,1, is rotated about the line y=2.
Solution
Mathematical Solution
According to Table 5.5.1, the "formula" 2 π ∫abρ ⅆs becomes
2 π ∫012−x21+ⅆⅆ x x22 ⅆx
=2 π ∫012−x21+4 x2 ⅆx
=2π3332ln2+23325+3364ln12+145
≐14.77521436
Maple Solution
In Figure 5.5.2(a) the tutor has been applied to the graph of y=x2 rotated about the line y=2. 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 5.5.2(a-b), the unevaluated integral, or the value of the surface area.
Figure 5.5.2(a) Surface of Revolution tutor
Student:-Calculus1:-SurfaceOfRevolution(x^2,x=0..1,distancefromaxis=2,partition=3,showsum=true, showsurface=false,output= plot,caption="",axes=frame,orientation=[-105,80,0],labels=[x,z,y],tickmarks=[2,[0],3],sumsurfaceoptions=[color=red]);
Figure 5.5.2(b) Segmentation into frustums
Figure 5.5.2(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 5.5.2(b), the tutor also provides the following midpoint Riemann approximating sum.
π∑i=131913i-132-2+19i2-24i2-4i+10≐14.59535361
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∫012− x21+ⅆⅆ x x22 ⅆx
2π3332ln2+23325+3364ln12+145
→at 10 digits
14.77521436
Application of the SurfaceOfRevolution command
Tools≻Load Package: Student Calculus 1
Loading Student:-Calculus1
SurfaceOfRevolutionx2,x=0..1,distancefromaxis=2, output=integral
∫01−2πx2−24x2+1ⅆx
SurfaceOfRevolutionx2,x=0..1,distancefromaxis=2
3316πln2+2316π5+3332πln12+145
<< Previous Example Section 5.5 Next Example >>
© Maplesoft, a division of Waterloo Maple Inc., 2024. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
Download Help Document
