(9/30/08)Math 10B. Lecture Examples.Section 8.2. Applications to geometry†Example 1 The region bounded by the curve y = x − x4and the x-axis is rotated aboutthe x-axis. Find the volume of the solid that is gene rated.Answer: Figures A1a and Figure A1b • [Volume] =19πx1y0.5−0.5y = x − x4Radius at xxr = x − x4The region that is rotated The cross secti on at xFigure A1a Figure A1bExample 2 Find the volume of the solid that is generate d when the triangle betweenthe lines y =12x and y = 2x for 0 ≤ x ≤ 2 is rotated about the x-axis.Answer: Figures A2a and A2b • [Volume] = 10π.x2y24y = 2xy =12xx12x2xThe triangle that is rotated The cross secti on at xFigure A2a Figure A2b†Lecture notes to accompany Section 8.2 of Calculus by Hughes-Hallett et al1Math 10B. Lecture Examples. (9/30/08) Section 8.2, p. 2Example 3 Find the length of y =23x3/2for 0 ≤ x ≤ 1.Answer: [Length] =23(23/2− 1)Example 4 Express the length o f y = exfor −1 ≤ x ≤ 1 as a definite integral.Answer: [Length] =Z1−1p1 + (ex)2dxInteractive ExamplesWork the following Interactive Examples on Shenk’s web page, http//www.math.ucsd.edu/˜ashenk/:‡Section 7.2: Examples 1–4Section 7.5: Example 1‡The chapter and section numbers on Shenk’s web site refer to his calculus manuscript and not to the chapters and sectionsof the textbook for the