Question 1

Suppose an object moves along a straight line with velocity $$v(t) = 3 \sin 2t (in m/s) $$for $$0 \leq t \leq 2\pi. $$What is the total distance traveled by the object on the interval $$[0, 2\pi]$$?

Accepted Answer

$$\displaystyle \$$int_0$$^{2\pi} |3 \sin 2t| \, dt = 12$$

Question 2

A block is attached to a spring and its velocity is given by $$v(t) = \frac{1}{4} \sin t (in m/s$$), with initial position $$s(0) = -\frac{1}{4} m. $$What is the position function $$s(t)$$ for $$t \geq 0$$?

Accepted Answer

$$s(t) = -\frac{1}{4} \cos t$$

Question 3

A car moves along a straight road with velocity $$v(t) = 6 - 2t (in ft/s) $$for $$0 \leq t \leq 6. $$What is the displacement of the car over this interval?

Accepted Answer

$$\displaystyle \$$int_0$$^6 (6 - 2t) \, dt = -18 ft$$

Question 4

A skydiver jumps from a helicopter and falls with a velocity that is $$80 m/s $$for $$0 \leq t \u003c 19 s$$, then decreases linearly to $$6 m/s $$over the next $$2$$ seconds, and remains at $$6 m/s $$until $$t = 40 s. $$What integral expression gives the total distance fallen before reaching the ground?

Accepted Answer

$$\displaystyle \$$int_0$$^{19} 80 \, dt + \int_{19}^{21} (783 - 37t) \, dt + \int_{21}^{40} 6 \, dt$$

Question 5

Given the acceleration $$a(t) = -9.8 m/s$^2$$ and initial velocity $$v(0) = 20 m/s$$, what is the velocity function $$v(t)$$?

Accepted Answer

$$v(t) = 20 - 9.8t$$

Question 6

A population of bacteria grows at a rate $$N'(t) = 90 e$$^{-0.1t}$$ cells/hr, with N(0) = 100$. What is the population function N(t$$)$$?

Accepted Answer

$$N(t) = 1000 - 900 e$$^{-0.1t}$$

Question 7

A book publisher's marginal cost function is $$C'(x) = 12 - 0.0002x$$ for $$0 \leq x \leq 50,000. $$What is the additional cost to increase production from $$12,000 to$$ $$15,000$$ books?

Accepted Answer

$$\displaystyle \int_{12000}^{15000} (12 - 0.0002x) \, dx$$

Question 8

Suppose the power consumption of a city is $$P(t) = 300 - 200 \sin \left(\frac{\pi t}{12}\right) MW $$for $$0 \leq t \leq 24$$ hours. What is the total energy consumed in one day (in megawatt-hours)?

Accepted Answer

$$\displaystyle \$$int_0$$^{24} P(t) \, dt = 7200$$ MWh

Question 9

If $$Q'(t) is $$the rate at which water flows into a tank and $$Q(0) = 0$$, which of the following expressions gives the total amount of water in the tank at time $$t$$?

Accepted Answer

$$Q(t) = Q(0) + \$$int_0$$^t Q'(x) \, dx$$

Question 10

A cyclist rides with velocity $$v(t) = 400 - 20t (in m/$$min) for $$0 \leq t \leq 10. $$How far does the cyclist travel in the first 5 minutes?

Accepted Answer

$$\displaystyle \$$int_0$$^5 (400 - 20t) \, dt = 1500 m$$

Question 11

Given the velocity $$v(t) = 3t^2$ - 6t on$$ $$[0,3]$$, what is the total distance traveled by the object?

Accepted Answer

$$\displaystyle \$$int_0$$^1 (3t^2$ - 6t) \, dt + \$$int_1$$^3 |3t^2$ - 6t| \, dt = \frac{16}{3}$$

Question 12

A tank is being filled at a rate $$Q'(t) = 3\sqrt{t} L/$$min. How much water flows into the tank in the first hour?

Accepted Answer

$$\displaystyle \$$int_0$$^{60} 3\sqrt{t} \, dt = 3 \cdot \frac{2}{3} t$$^{3/2}$$ \Big|_0$$^{60}$$ = 2 \cdot 60$$^{3/2}$$ L

Question 13

If the velocity of an object is always positive on an interval $$[a, b]$$, which of the following statements is true?

Accepted Answer

The displacement and the distance traveled are equal on $$[a, b].$$

Question 14

A mass on a spring moves with velocity $$v(t) = 2\pi \cos(\pi t)$$, $$s(0) = 0. $$What is the position function $$s(t)$$?

Accepted Answer

$$s(t) = 2 \sin(\pi t)$$

Question 15

Suppose the rate of change of a chemical in a reaction is $$Q'(t) = -0.5 Q(t)$$ with $$Q(0) = 100. $$What is the expression for $$Q(t)$$?

Accepted Answer

$$Q(t) = 100 e$$^{-0.5 t}$$

Question 16

A train decelerates with $$a(t) = -1280 (1 + 8t$$)^{-3}$$ mi/hr^2$ from v(0) = 80$ mi/hr. What is the velocity function v(t$$)$$?

Accepted Answer

$$v(t) = 80 + \$$int_0$$^t -1280 (1 + 8x$$)^{-3}$$ dx$$

Question 17

If $$v(t) is $$the velocity of an object and $$v(t)$$ changes sign on $$[a, b]$$, which of the following is true?

Accepted Answer

The distance traveled is greater than or equal to the magnitude of the displacement.

Applications of Integration: Velocity, Net Change, and Physical Applications

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