Math 90 (Calculus II) Final Review. Sketch the region enclosed by the given curves and find the area of the region. a. y = 7 x, y = x + 4 b. y = cos ( πx ), y = x. Use the specified method to find the volume generated by rotating the region bounded by the given curves and axis of rotation. a. Use the Disk/Washer Method: y = 4 x, y = x, About x axis b. Use the Shell Method: y = x x, y = x, About the line x = 3. a. Find the Arc Length of the graph of the function over the indicated interval. y = x7 4 +, [, ] 0x5 b. Find the Surface Area generated by revolving the curve on the indicated interval about the y-axis. y = 9 x, 0 x 3 4. Use integration to solve the following application problems involving Work, Centroids, Fluid Pressure, and Fluid Force. a. Eighteen foot-pounds of work is required to stretch a spring 4 inches from its natural length. Find the Work required to stretch the spring an additional 3 inches. b. Find M x, M y, and (x, y) for the lamina of uniform density ρ bounded by the graphs of the equations. y = x +, y = 3 x +
c. Find the Fluid Force on the vertical side of the triangular water-filled tank, where the dimensions are given in feet. 4 ft 3 ft d. Find the Fluid Force on the rectangular vertical plate submerged in water, where the dimensions are given in meters and the weight-density of water is 9800 N m 3. m 5m 5 plate water m 5. Evaluate the following Integrals a. t 3 t 4 + dt b. 3x x+4 dx c. e 4x cos(x)dx d. lnx x 3 dx e. sin 7 (x)cos (x) dx f. sec 5 x tan 3 x dx g. x3 dx x 5 h. 5x +4 dx x 4 i. x3 x+3 x +x dx j. x 6x+ x 3 +x +x dx
6. Determine whether the improper integral converges or diverges. Evaluate the integral if it converges. a. 8 3 0 8 x dx b. 3 3 dx x 7. Test the following series for convergence. Show your work and state which test you are using. a. (n+)! n= b. e n 5n! n= ln(n+) c. n= d. n+ n= 5 n 3 e. n= f. n(n+3) ( ) n n n= n 3 5 g. ( n n= h. 5n+ )n ( ) n n n=0 n! 8. Find the interval of convergence for each power series. Be sure to check for convergence at the endpoints. a. ( )n+ x n n=0 6 n b. ( )n+ (x ) n+ n + n=0
9. Solve the following applications of geometric series, power series, and/or Taylor/Maclaurin series. a. Find a geometric power series for the function centered at c = 0 and determine the interval of convergence. f(x) = 5 4 x b. Use integration to find a power series for the function centered at c = 0 and determine the interval of convergence. f(x) = ln(x + ) = x+ dx c. Use differentiation to find a power series for the function centered at c = 0 and determine the interval of convergence. f(x) = dx = d [ ] (x+) dx x+ 0. Find the center, foci, vertice(s), eccentricity, directrix, and asymptotes as appropriate for the given conic sections. Graph the conic section. Hint: Remember standard form! a. x + 4x + 4y 4 = 0 b. x + 0y 6x + 0y + 8 = 0 c. 9x y 36x 6y + 8 = 0. Eliminate the parameter to write the corresponding rectangular equation. Sketch the curve represented by the parametric equations. x = 8 cos θ and y = 8 sin θ. Find dy dx and d y. Find the slope and concavity (if possible) at the given value of the dx parameter. x = t + 5t + 4, y = 4t, t = 0
3. Find all points (if any) of horizontal and vertical tangency to the curve. x = cos θ, y = sin θ 4. Perform the following conversions between rectangular and polar form. a. Find the corresponding rectangular coordinates for the given polar coordinates. ( 4, 3π 4 ). b. Given the rectangular coordinate for a point (, ), find two sets of polar coordinates of the point for 0 θ < π. c. Convert the rectangular equation to polar form and describe what the graph would look like. x y = 9 d. Convert the polar equation to a rectangular equation and describe what the graph would look like. r = 3 sin θ 5. a. Sketch a graph of the polar equation r = 4( sin θ) and find the tangent line(s) of the curve at θ = π. b. Sketch a graph of the polar equation r = 3 cos θ. 6. a. Find the area of two petals of the curve r = 4 sin 3θ. b. Find the area inside of r = cos θ and outside of r =. 7. Find the length of the curve over the given interval. r = 4 sin θ, [0, π]
a. 4 b. a. 3a. 08 5 π b. π 6 ANSWER KEY 4 3 4 π 0539 40 9.68 3b. π 6 ( 373 ) 7.39 4a. 37.5 ft lbs 4b. M x = 45 ρ, M 4 y = 8 ρ, (8, 5 ) 5 5 4c. 374.4 lbs 4d. 7,500 Newtons 5a. 5c. 6 (t4 + ) 3 + C 5b. 3x ln x + 4 + C (by Integration Rules) 5 e4x cos x + 0 e4x sin x + C 5d. x ln x + C 4x (by Parts) 5e. 6 sin8 x + C 5f. sec 7 x 7 sec5 x 5 + C (by Trig Integrals) 5g. 3 x 5 (x + 50) + C 5h. (5x +4) 3 + C (by Trig Substitution) x 3 5i. x x + ln x + x + C 5j. ln x x+ + 9 x+ + C (by Partial Fractions) 6a. Converges, 6b. Diverges 7a. Diverges by the nth term test 7b. Converges by Geometric Series ( r = e < ) OR by Integral test 7c. Diverges by the Integral Test 7d. Diverges by p Series ( p = 5 < ) 7e. Converges Telescoping series 7f. Alternating Series converges absolutely by limit comparison to a p Series. 7g. Converges by the Root Test ( 5 < ) 7h. Converges by the Ratio Test (0 < ) 8a. Converges for 6 < x < 6 8b. Converges for 0 < x 9a. 5 xn n=0 (, ) 9b. n=0 (, ] 4 n+ 9c. ( ) n+ (n + )x n (, ) n=0 ( ) n x n+ n+
#0 Type Center Foci Vertice(s) Eccent. Directrix Asympt a Parabola NA (-, ) (-, ) NA y = 3 NA b Ellipse (3, -) (3 + 3 0, ) 3 (3 0, ) (4, -) (, -) 3 NA NA 0 c Hyperbola (, -3) ( + 0, 3) ( 0, 3) (, -3) (3, -3) 0 NA y = 3x-9 y = -3x+3. x + y = 64, Circle centered at origin with radius of 8.. dy = 4, d y = dx t+5 dx 8 (t+5) 3, Slope = 4 5, Concavity = 8 5 (Concave Down) 3. Horiz. Tang. at θ = π 4, 3π 4, 5π 4, 7π 4 (, ), ( Vertical Tangents at θ = 0, π (, 0), (, 0), ), (, ), (, ) 4a. (, ) 4b. (4, 5π 4 ) and ( 4, π 4 ) 4c. r cos θ = 9 (Hyperbola) 4d. x + (y 3 ) = 9 4 (Circle) 5a. 5b. No tangent at pole. 6. a. Area: A = 8π 3 6b. Area: A = π 3 + 3 7. Length: s = 4π