Question 1

An object moves along the x-axis with velocity $$v(t)$$, and its position at $$t=0 is$$ $$s(0) = 40. If $$the displacement from $$t=0 to$$ $$t=7 is$$ $$3.5$$, what is the position $$s(7)$$?

Accepted Answer

$$s(7) = 43.5$$

Question 2

Given the velocity function $$v(t) of an $$object, the displacement from $$t=0 to$$ $$t=7 is $$given by which of the following integrals?

Accepted Answer

$$\displaystyle \$$int_0$$^7 v(t)\,dt$$

Question 3

If the total distance traveled by an object from $$t=0 to$$ $$t=7 is$$ $$6.5$$, but the displacement is $$3.5$$, what does this indicate about the motion of the object?

Accepted Answer

The object moved in the negative direction at some point.

Question 4

Which of the following integrals correctly represents the area of the region $$R$$ bounded by $$y = (x-2)^2$, y = 9$, and x = 3$, for x$$ \geq 3$$?

Accepted Answer

$$\displaystyle \$$int_3$$^5 9 - (x-2)^2$\,dx$$

Question 5

Suppose $$y = (x-2)^2$ and y = 9$ intersect at x = -1$ and x = 5$. What is the length of the interval over which you integrate to find the area between these curves for x$$ \geq 3$$?

Accepted Answer

$$4$$

Question 6

Express the area of the region $$R$$ bounded by $$y = (x-2)^2$, y = 9$, and x = 3$ as an integral with respect to y$. Which of the following is correct?

Accepted Answer

$$\displaystyle \$$int_1$$^9 (2 + \sqrt{y}) - 3\,dy$$

Question 7

Given the region $$R$$ bounded by $$y = 2 + \sqrt{x-3}$$, $$y = -2x + 18$$, and $$y = -\frac{1}{3}x + 3$$, which of the following correctly expresses the area of $$R as a $$sum of integrals with respect to $$x$$?

Accepted Answer

$$\displaystyle \$$int_3$$^7 (2 + \sqrt{x-3}) - (-\frac{1}{3}x + 3)\,dx + \$$int_7$$^9 (-2x + 18) - (-\frac{1}{3}x + 3)\,dx$$

Question 8

Convert $$y = -2x + 18 to$$ $$x as a $$function of $$y$$ for use in integrating with respect to $$y.$$

Accepted Answer

$$x = 9 - \frac{1}{2}y$$

Question 9

For the region bounded by $$y = 2 + \sqrt{x-3}$$, $$y = -2x + 18$$, and $$y = -\frac{1}{3}x + 3$$, which of the following expresses the area as a sum of integrals with respect to $$y$$?

Accepted Answer

$$\displaystyle \$$int_0$$^2 (9 - \frac{1}{2}y) - (9 - 3y)\,dy + \$$int_2$$^4 (9 - \frac{1}{2}y) - (3 + (y-2)^2$)\,dy$$

Question 10

A car moves along a straight road with velocity $$v(t). If$$ $$v(t) is $$negative for some interval, what does this mean about the car's motion?

Accepted Answer

The car is moving backward.

Question 11

Suppose $$v(t) is $$the velocity of a particle. Which integral gives the total distance traveled from $$t=0 to$$ $$t=7$$?

Accepted Answer

$$\displaystyle \$$int_0$$^7 |v(t)|\,dt$$

Question 12

If $$s(0) = 40$$ and the object’s velocity is negative for $$1$$ unit of time with $$v(t) = -1$$ during that interval, how does this affect the position $$s(t)$$?

Accepted Answer

The position decreases by $$1.$$

Question 13

Given $$y = (x-2)^2$, solve for x$ in terms of y$ for use in integrating with respect to y$.

Accepted Answer

$$x = 2 + \sqrt{y} or$$ $$x = 2 - \sqrt{y}$$

Question 14

A region is bounded above by $$y = 9$$ and below by $$y = (x-2)^2$ for x$ in $$[3,5]$$. What is the area of this region?

Accepted Answer

$$\displaystyle \$$int_3$$^5 9 - (x-2)^2$\,dx$$

Question 15

If $$v(t) is $$the velocity of a particle and $$\$$int_0$$^7 v(t)\,dt = 3.5$$, what does this value represent?

Accepted Answer

The net change in position (displacement) of the particle.

Question 16

Suppose you want to find the area between $$y = -2x + 18$$ and $$y = -\frac{1}{3}x + 3$$ from $$x = 7 to$$ $$x = 9. $$Which integral would you use?

Accepted Answer

$$\displaystyle \$$int_7$$^9 (-2x + 18) - (-\frac{1}{3}x + 3)\,dx$$

Question 17

If $$y = 2 + \sqrt{x-3}$$, solve for $$x in $$terms of $$y$$ for use in integrating with respect to $$y.$$

Accepted Answer

$$x = 3 + (y-2)^2$

Applications of Integration: Displacement, Distance, and Area Between Curves

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