Question

Let C be the portion of the curve rve y = x3 + 3x – 1 between the points (1, 3) and (2, 13). (a) Write down a definite integral for the arclength of C. Do not evaluate the integral. (b) Write down a definite integral for the surface area of the surface of revolution obtained by revolving Cabout the x-axis. Do not evaluate the integral. (c) Write down a definite integral for the surface area of the surface of revolution obtained by revolving C about the y-axis. Do not evaluate the integral.

Answer 1

portion of the curne Given that c is the y=x432 -1 between the points (1,3) and (2,13) @ It is asked to write down an integral (not to evaluate) ters, the are lengin of c. • No te det C be a Then the length such that wine y=fa), alxeb of the une C is given by the integral L b SJ1+ (4'))² de a . We da hane y = f(a)=2+32-1 - ')=d(2457-13 - 33+3 observe that exca 2 (3x43)2 de 4. length,

b) It ý asked to write down an integral for the surface area of the surface of revolutień cbtained by revolving c about the a-anis. Note het C be a y=fQ) cine such that a£x Eb, then the of see e face of Q cao ared about a - arus is, b fG) / 1 + ( f'G.) ² de Ag = am a The area of surface of pevoluteći arus is of the come about .b Ay= 25 Xit capit (8!@))? dax Va

2 Ax= an fee Ax= a (x² + 3x-1) [1+ (3 x 4 3 ² . one (C It is asked to write down a depirate integral for the sultace a sea of the surface of revolution obtau'red revolving c about the y-axis by 2 Ay . at x √ it (3x²+3)2 dhe