Math 1172 Sample Test #1 1. Find the area of the region enclosed by the curve y = x 2 + 1 and the line y = 2x + 1 € 2. € Find the area of the region enclosed by the graphs of x = y 2 and y = x − 2 3. 2. € € 1. -1. 1. 2. 3. 4. 5. -1. -2. -3. 3. Find the length of the curve y = x 4 x −2 + , 1≤ x ≤ 2 8 4 ⎛ dy ⎞ 2 (Hint: 1+ ⎜ ⎟ is a perfect square.) ⎝ dx ⎠ 4. € Find the length of the curve y = 2(x + 1) 3 / 2 , 0 ≤ x ≤ 1 € 5. Find the area of the surface obtained by x 4 x −2 + , 1 ≤ x ≤ 2 about the x-axis. (a) rotating the € curve y = 8 4 x 4 x −2 + , 1 ≤ x ≤ 2 about the y-axis. (b) rotating the curve y = 8 4 € € Math 1172 Sample Test #1 Page 2 Use this sketch for problems 6 and 7. 6. Use the washer method to find the volume of the solid generated by rotating the region bounded by the curve y = x2 and the line y = x about the x-axis. 7. Use the cylindrical shell method to find the volume of the solid generated by rotating the region bounded by the curve y = x2 and the line y = x about the y-axis. 8. Find the volume of the solid generated by rotating the region bounded by the curve x = y2 + 1 and the line x = 2y + 1 about the x-axis. 9. A spring has a natural length of 0.3m. It takes a force of 10N to keep the spring stretched to a length of 0.35m. Find the work done in stretching the spring from its natural length to a length of 0.4m. Math 1172 Sample Test #1 Page 3 For problems 10 and 11, use ρ = 1000 kg/m3 and g = 9.8 m/s2 10. A cylindrical tank of radius 2 meters and height 5 meters is positioned on a tower so that the bottom of the tank is 20 meters above the ground. Find the work done to fill € pumped up from € ground level. the tank with water 11. The lower edge of a dam is defined by the parabola y = x 2 /4, −10 ≤ x ≤ 10 . The upper edge of the dam is defined by the line y = 25 . (Lengths are measured in meters.) Find the force on the dam if the water level is at the top of the dam. € 12. The population of a town increases from 30,000 in 1990 to 40,000 in 2005. Devise € the exponential growth function that fits this data and use it to predict when the population will reach 50,000. 13. A drug is eliminated from a body at a rate of 12% per hour. Devise the exponential growth function that fits this data and use it to predict when the amount of the drug will reach 20% of the initial dose. 14. Uranium-238 (U-238) has a half-life of 4.5 billion years. 75% of the original U-238 in a rock still remains. How old is the rock? 15. Find the indefinite integrals: dx (a) ∫ 1 − sin x (b) ∫ (c) ∫ sec(2x)tan(2x)dx (d) ∫x (e) ∫e (f) ∫x2 (g) ∫x € € € € x 2 +1 dx x +1 x +1 dx 2 +1 5x € € € 2 dx x2 dx dx − 6x + 25 Math 1172 Sample Test #1 Page 4 16. Evaluate the definite integrals. (a) ∫ (b) ∫ € 4 x 0 2 x +9 dx 1 0 dx 4 − x2 17. Use integration by parts to find the integrals. € (a) € € € € € € € ∫ x e dx 2 x (b) ∫x (c) ∫ xe 5x (d) ∫x 2 (e) ∫ x cos 3x dx (f) ∫ ln x dx (g) ∫ arctan x dx 2 ln x dx dx cos x dx