Chapter 5 | Integration
575
5.4 EXERCISES Use basic integration formulas to compute the following antiderivatives or definite integrals. 207. ∫ ⎛ ⎝ x − 1 x ⎞ ⎠ dx 208. ⌠ ⌡ ⎛ ⎝ e 2 x − 1 2 e x /2 ⎞ ⎠ dx 209. ⌠ ⌡
220. Write an integral that quantifies the increase in the volume of a cube when the side length doubles from s unit to 2 s units and evaluate the integral. 221. Write an integral that quantifies the increase in the surface area of a sphere as its radius doubles from R unit to 2 R units and evaluate the integral. 222. Write an integral that quantifies the increase in the volume of a sphere as its radius doubles from R unit to 2 R units and evaluate the integral. 223. Suppose that a particle moves along a straight line with velocity v ( t ) =4−2 t , where 0≤ t ≤2 (inmeters per second). Find the displacement at time t and the total distance traveled up to t =2. 224. Suppose that a particle moves along a straight line with velocity defined by v ( t ) = t 2 −3 t −18, where 0≤ t ≤6 (in meters per second). Find the displacement at time t and the total distance traveled up to t =6. 225. Suppose that a particle moves along a straight line with velocity defined by v ( t ) = | 2 t −6 | , where 0≤ t ≤6 (in meters per second). Find the displacement at time t and the total distance traveled up to t =6. 226. Suppose that a particle moves along a straight line with acceleration defined by a ( t ) = t −3, where 0≤ t ≤6 (in meters per second). Find the velocity and displacement at time t and the total distance traveled up to t =6 if v (0) =3 and d (0) =0. 227. A ball is thrown upward from a height of 1.5 m at an initial speed of 40 m/sec. Acceleration resulting from gravity is −9.8 m/sec 2 . Neglecting air resistance, solve for the velocity v ( t ) and the height h ( t ) of the ball t seconds after it is thrown and before it returns to the ground. 228. A ball is thrown upward from a height of 3 m at an initial speed of 60 m/sec. Acceleration resulting from gravity is −9.8 m/sec 2 . Neglecting air resistance, solve for the velocity v ( t ) and the height h ( t ) of the ball t seconds after it is thrown and before it returns to the ground. 229. The area A ( t ) of a circular shape is growing at a constant rate. If the area increases from 4 π units to 9 π units between times t =2 and t =3, find the net change in the radius during that time.
dx 2 x
210. ⌠ ⌡
x −1 x 2
dx
π (sin x −cos x ) dx
211. ∫
0
π /2
212. ∫
( x −sin x ) dx
0
213. Write an integral that expresses the increase in the perimeter P ( s ) of a square when its side length s increases from 2 units to 4 units and evaluate the integral. 214. Write an integral that quantifies the change in the area A ( s ) = s 2 of a square when the side length doubles from S units to 2 S units and evaluate the integral. 215. A regular N -gon (an N -sided polygon with sides that have equal length s , such as a pentagon or hexagon) has perimeter Ns . Write an integral that expresses the increase in perimeter of a regular N -gon when the length of each side increases from 1 unit to 2 units and evaluate the integral. 216. The area of a regular pentagon with side length a >0 is pa 2 with p = 1 4 5+ 5+2 5. The Pentagon in Washington, DC, has inner sides of length 360 ft and outer sides of length 920 ft. Write an integral to express the area of the roof of the Pentagon according to these dimensions and evaluate this area. 217. A dodecahedron is a Platonic solid with a surface that consists of 12 pentagons, each of equal area. By how much does the surface area of a dodecahedron increase as the side length of each pentagon doubles from 1 unit to 2 units? 218. An icosahedron is a Platonic solid with a surface that consists of 20 equilateral triangles. By how much does the surface area of an icosahedron increase as the side length of each triangle doubles from a unit to 2 a units? 219. Write an integral that quantifies the change in the area of the surface of a cube when its side length doubles from s unit to 2 s units and evaluate the integral.
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