Answer: Find the volume of the solid generated by revolving the shaded region about the given axis. 2) About the x-axis. y = 9 - x π.

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1 Final Review Study All Eams. Omit the following sections: 6.,.6,., 8. For Ch9 and, study Eam4 and Eam 4 review sheets. Find the volume of the described solid. ) The base of the solid is the disk + y 4. The cross sections by planes perpendicular to the y-ais between y = - and y = are isosceles right triangles with one leg in the disk. 64 Find the volume of the solid generated by revolving the shaded region about the given ais. ) About the -ais y 4 y = 9-6 π Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the -ais. ) y = +, y =, =, = 4π 4) y =, y =, =, = π ) y =, y =, = 4/π Find the volume of the solid generated by revolving the region about the given line. 6) The region in the first quadrant bounded above by the line y =, below by the curve y =, and on the left by the y-ais, about the line y = 9 π

2 ) The region in the first quadrant bounded above by the line 6 + y =, below by the -ais, and on the left by the y-ais, about the line = - 64π Find the volume of the solid generated by revolving the region about the y-ais. 8) The region enclosed by = y/, =, y = 8 96 π 9) The region in the first quadrant bounded on the left by y = 6, on the right by the line = 6, and above by the line y = 8π Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated ais. ) About the -ais y y = 4 = 4 4 = 6 - y 4 64 π Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the y-ais. ) y = 4, y = 4, for 6 π Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the -ais. ) y = 6, y =, y = 6 π

3 Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves about the given lines. ) y =, y =, = ; revolve about the -ais π Find the length of the curve. 4) = (y - ) / from y = 9 to y = 6 4 ) y = t4 - dt, - y Find the area of the surface generated by revolving the curve about the indicated ais. 6) y = 6 -,..; -ais 6π Find the value of df-/d at = f(a). ) f() = - - 4, 8, a = - 6 Find the derivative of y with respect to, t, or θ, as appropriate. 8) y = ln (cos (ln θ)) - tan (ln θ) θ Evaluate the integral. d 9) + 8 ln 8 ln + 8 ln + C ) sec tan -4 + sec d ln -4 + sec + C

4 Use logarithmic differentiation to find the derivative of y. ) y = sin + 8 sin + 8 cot + ( + 8) Solve the problem. ) Find the volume of the solid that is generated by revolving the area bounded by the -ais, the curve y = 9, =, and = about the -ais. + 9 π ln Find the derivative of y with respect to, t, or θ, as appropriate. ) y = eθ(sin θ - cos θ) 4eθ sin θ 4) y = esin t (ln t + ) esin t (cos t)(ln t + ) + t Find dy d. ) e = sin ( + y) e - cos ( + y) cos ( + y) Evaluate the integral. 6) e/ d - e / + C ) π/ ( + etan ) sec d e 4

5 Solve the initial value problem. 8) dy d = e 4 cos e4, y() = y = 4 sin e 4-4 sin Find the derivative of y with respect to the independent variable. 9) y =log 6 + ln - (+) Evaluate the integral. ) log9 d (ln ) ln 9 + C ) ( 6 + ) d Find the length of the curve. ) y = (e + e-) from = to = e - e Use l'hopital's Rule to evaluate the limit. ) lim cos - - Use l'ho^pital's rule to find the limit. 4) lim sin

6 Find the limit. ) lim + -/ln e L'Hopital's rule does not help with the given limit. Find the limit some other way. 6) lim cot cos Evaluate the integral by using a substitution prior to integration by parts. ) e + 6 d e + 6 [ ] + C Evaluate the integral. The integral may not require integration by parts. 8) 4e d e + C Solve the problem. 9) Find the area of the region enclosed by y = sin and the -ais for π. π 4) Find the volume of the solid generated by revolving the region bounded by the curve y = cos and the -ais, π π, about the -ais. π Use integration by parts to establish a reduction formula for the integral. 4) n e-a d n e-a d = - n e-a a + n a n- e-a d Use any method to evaluate the integral. 4) csc tan d - csc + C 6

7 4) sin d 4-4 sin - 8 cos + C Use a trigonometric substitution to evaluate the integral. 44) e d - e sin- (e) + C 4) d + tan- + C Evaluate the improper integral or state that it is divergent. d 46) ( - 9)( - 8) 4) ln e-6 d - Evaluate the improper integral. 48) 9 - d Determine whether the improper integral converges or diverges. 49) - -4/ d Diverges ) + Diverges ) e- sin d Converges