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Math 172 Exam 1 Review Feb 2016 The problems on this page form rough approximations of an exam. 1. Integrate. (Sunday) Z 3 √ (a) t2 1 + t dt Z (b) 0 2. Integrate. (Monday) Z eπ sin (ln t) (a) dt t 1 Z (b) Z dy y2 + 4 (c) x arctan x dx Z π/2 sin3 2θ dθ (d) 0 Z dy p 2 − y2 (c) x ln x dx Z (d) π/4 sec4 θ dθ 0 3. Consider the area between the graphs of the functions y = sin x and y = cos x for x ∈ [0, π/2]. Carefully sketch the graphs and find the area. 4. Find the volume of the solid with base bounded by y = |2x| and y = 2 with cross sections perpendicular to the y-axis that are squares. 5. Consider the region between the graph of y = ex and the x-axis for x ∈ [0, 1]. Carefully sketch the region. Find the volume of the solid generated by rotating the region about the y-axis. 6. Consider the region bounded by the graphs of y = x2 and y = x + 2. Carefully sketch the region. Find the volume of the solid generated by rotating the region about the line y = −1. 7. Calculate the work (in joules) required to pump all of the mystery fluid of density ρ out of a full tank on the planet Zanabar with gravitational constant g. The tank is in the shape of the graph of y = x4 for x ∈ [0, 2] rotated about the y-axis. The mystery fluid is pumped out a spout of length 1 m from the top of the tank. 8. Integrate. (Sunday) Z t dt √ (a) 4 − t2 9. Integrate. (Monday) Z √ (a) t 2 − t dt Z 1/4 (b) 1/8 Z (b) 0 1/3 dy p 1 − 16y 2 dy 2 9y + 1 Z π/4 x sec2 x dx (c) Z sin4 θ dθ Z sec4 θ tan θ dθ (d) 0 Z (c) x2 sin 2x dx (d) 10. Consider the area between the graphs of the functions y = x2 and y = x for x ∈ [0, 2]. Carefully sketch the graphs and find the area. 11. Find the volume of the solid with base bounded by y = sections perpendicular to the x-axis that are semicircles. 1 x and x-axis for x ∈ [1, 3] with cross 12. Consider the region bounded by the graphs of y = x2 , y = 12 − 4x, and the x-axis. Express the volume of the solid generated by rotating the region about the x-axis as an integral using both the disk/washer/slicing method and the shell method. Do not evaluate either integral. 13. Consider the region bounded by the graphs of x = y 2 and x = y + 2. Carefully sketch the region. Find the volume of the solid generated by rotating the region about the line y = 2. 14. Calculate the work (in joules) required to pump all of the mystery fluid of density ρ out of a full tank on the planet Zanabar with gravitational constant g. The tank is a horizontal cylinder of radius r and length l. The mystery fluid is pumped out of the top of the tank. Additional review problems. 1. Integrate. Z e ln x (a) dx x 1 Z (e) e2x sin 3x dx Z (b) 0 3 Z 1 Z π/2 1+x 1 + 2x √ dx (c) dx (d) x cos x dx 9 + x2 1 − x2 0 0 Z Z (f) sin x cos x sin 2x dx (g) sec3 x tan3 x dx 2. Consider the area between the graphs of the functions y = tan x and y = sec x for x ∈ [0, π/4]. Carefully sketch the graphs and find the area. 3. Find the volume of the solid with base bounded by y = tan x and x-axis for x ∈ [0, π/4] with cross sections perpendicular to the x-axis that are squares. 4. Consider the region bounded by the graph of y = sin (x/2) and the x-axis for x ∈ [0, 2π]. (a) (b) (c) (d) Find Find Find Find the the the the volume volume volume volume of of of of the the the the solid solid solid solid generated generated generated generated by by by by rotating rotating rotating rotating the the the the region about the x-axis. region about the y-axis. region about the line y = 1. region about the line x = −1. √ 5. The region in the first quadrant bounded by the graphs of y = x and y = 2 − x, the shaded region in Figure 1 below, is revolved around the x-axis. Express the volume of the resulting solid as an integral, or integrals, using the Disk Method and the Shell Method. Evaluate the easier of the two. 6. The region in the first quadrant bounded by the graphs of y = 5 − x2 and y = 6/x − 2, the shaded region in Figure 2 below, is revolved around the line x = −1. Express the volume of the resulting solid as an integral using the Disk Method and the Shell Method. Evaluate the easier of the two. 4 1 3 2 1 1 2 -1 Figure 1: 0 1 2 Figure 2: 7. Show the volume of a sphere of radius R is V = 43 πR3 . 8. Calculate the work (in joules) to build a pyramid with square base of side length s and height h out of a material of density ρ with a gravitational constant g. 9. Let’s build a snowman! (a) Calculate the work (in joules) to buld a sphere of radius r with center at height h out of snow of density ρ with a gravitational constant g. (b) Using part (a) find the work to build a snowman if the base is a sphere of radius 1/2 m, the middle sphere has radius 1/3 m, and the top sphere has radius 1/4 m. (c) Adding a top hat, coal for eyes, and a carrot nose is up to you.