386Chapter 7 Applications of Definite IntegralsIn Exercises 1–8, the function v(t) is the velocity in msec of aparticle moving along the x-axis. Use analytic methods to do each ofthe following:(a)Determine when the particle is moving to the right, to theleft, and stopped.(b)Find the particle’s displacement for the given time interval. Ifs(0)3, what is the particle’s final position?(c)Find the total distance traveled by the particle.1.v t5 cost,0t2pSee page 389.2.v t6 sin 3t,0tp2See page 389.3.v t499.8t,0t10See page 389.4.v t6t218t12,0t2See page 389.5.v t5 sin2tcost,0t2pSee page 389.6.v t4t,0t4See page 389.7.v tesintcost,0t2pSee page 389.8.v t1tt2,0t3See page 389.9.An automobile accelerates from rest at 13tmphsec for 9 seconds.(a)What is its velocity after 9 seconds?63 mph(b)How far does it travel in those 9 seconds?344.52 feet10.A particle travels with velocityv tt2 sintm secfor 0t4 sec.(a)What is the particle’s displacement?1.44952 meters(b)What is the total distance traveled?1.91411 meters11.ProjectileRecall that the acceleration due to Earth’s gravity is32 ftsec2. From ground level, a projectile is fired straightupward with velocity 90 feet per second.(a)What is its velocity after 3 seconds?6 ft/sec(b)When does it hit the ground?5.625 sec(c)When it hits the ground, what is the net distance it hastraveled?0(d)When it hits the ground, what is the total distance it hastraveled?253.125 feetIn Exercises 12–16, a particle moves along the x-axis (units in cm).Its initial position at t0 sec is x015. The figure shows thegraph of the particle’s velocity v t. The numbers are the areasofthe enclosed regions.45abc24
12.What is the particle’s displacement between t0 and tc?13.What is the total distance traveled by the particle in the sametime period?33 cm14.Give the positions of the particle at times a,b, and c.15.Approximately where does the particle achieve its greatestpositive acceleration on the interval 0,b?ta16.Approximately where does the particle achieve its greatestpositive acceleration on the interval 0,c?t cIn Exercises 17–20, the graph of the velocity of a particle moving onthe x-axis is given. The particle starts at x2 when t0.
(a)Find where the particle is at the end of the trip.(b)Find the total distance traveled by the particle.18.104.22.168.21.U.S. Oil ConsumptionThe rate of consumption of oil in theUnited States during the 1980s (in billions of barrels per year)is modeled by the function C27.08•et25, where tis thenumber of years after January 1, 1980. Find the total consumptionof oil in the United States from January 1, 1980 to January 1,1990.