Deck 31: Applications of the Integral

Question 1

A particle undergoes an acceleration $a=2 t+8\left(\mathrm{~mm} / \mathrm{s}^{2}\right)$ and its velocity at $2 \mathrm{~s}$ is $15 \mathrm{~mm} / \mathrm{s}$ .
(a) Find the relation between $v$ and $t$ for the particle.
(b) Find the velocity of the particle after $5 \mathrm{~s}$ .

Answer 1

(a) $v=t^{2}+8 t-5$
(b) $60 \mathrm{~mm} / \mathrm{s}$

Question 2

The acceleration of a point is given by $a=5 t-t^{2}\left(\mathrm{~m} / \mathrm{s}^{2}\right)$ . Write an equation for the velocity, if the velocity is $4 \mathrm{~m} / \mathrm{s}$ at $2 \mathrm{~s}$ .

Answer 2

$\quad v=\frac{5}{2} t^{2}-\frac{1}{3} t^{3}-\frac{10}{3}$

Question 3

A particle starts from the origin. If $v_{x}=t^{2}-1$ and $v_{y}=5 t$ , find the distance (in $\mathrm{cm}$ ) between the particle and the origin at $3.00 \mathrm{~s}$ .

Answer 3

$23.3 \mathrm{~cm}$

Question 4

A point starts from $(6,2)$ with initial velocities $v_{x}=3 \mathrm{~cm} / \mathrm{s}$ and $v_{y}=2 \mathrm{~cm} / \mathrm{s}$ and moves along a curved path. If $a_{\mathrm{x}}=2 t$ and $a_{\mathrm{y}}=6 t$ , find $v_{\mathrm{x}}$ and $v_{\mathrm{y}}$ at $t=3 \mathrm{~s}$ .

Question 5

A gear starts from rest and accelerates at $9.12 t^{2} \mathrm{rad} / \mathrm{s}^{2}$ . Find the total number of revolutions after $10.0 \mathrm{~s}$ .

Question 6

A wheel starts from rest and accelerates at $4.10 \mathrm{t} \mathrm{rad} / \mathrm{s}^{2}$ . Find the angular velocity after $7.50 \mathrm{~s}$ .

Question 7

A wheel starts from rest and accelerates at $4.10 \mathrm{trad} / \mathrm{s}^{2}$ . Find the total number of revolutions after $7.50 \mathrm{~s}$ .

Question 8

An object starts at the position $(5,20)$ and has initial velocities $v_{x}=15 \mathrm{~m} / \mathrm{s}$ and $v_{y}=-5 \mathrm{~m} / \mathrm{s}$ . The equations for acceleration are $a_{x}=8 t-\frac{1}{2} t^{2}$ and $a_{y}=1-5 t$ . Find the equations for the $\mathrm{x}$ and $\mathrm{y}$ components of velocity and displacement.

Question 9

The angular acceleration of a bicycle wheel is given by $\alpha=4 \pi t$ . Find the equation for the angular velocity if the bicycle wheel has an angular velocity of $8 \mathrm{rad} / \mathrm{s}$ at $t=0.4 \mathrm{~s}$ .

Question 10

A car starts from rest with an acceleration $a=18 t-\frac{1}{2} t^{2}$ after 30 minutes.

Question 11

An object starts at the position $(5 \mathrm{~cm}, 16 \mathrm{~cm})$ and has the velocities $v_{x}=3 t-1$ and $v_{y}=5-2 t$ . Find the position of the object at $8 \mathrm{~s}$ . How far has the object travelled in this time?

Question 12

A gear has acceleration $\alpha=\frac{1}{\sqrt{t}}\left(\mathrm{rad} / \mathrm{s}^{2}\right)$ . If the gear has an initial rotational velocity of $\pi \mathrm{rad} / \mathrm{s}$ , find the equation for angular velocity.

Question 13

The current to a capacitor is given by $i=2 t^{2}-t$ . The initial charge on the capacitor is 5.13 coulombs. Find an expression for the charge on the capacitor.

Question 14

A 10.0-F capacitor has an initial voltage of $1.00 \mathrm{~V}$ . It is charged with a current given by $i=t \sqrt{t^{2}+16}$ . Find the voltage across the capacitor at $5.00 \mathrm{~s}$ .

Question 15

The voltage across a $1.51-\mathrm{H}$ inductor is $v=\sqrt{82 t} \mathrm{~V}$ . Find the current in the inductor at $2.50 \mathrm{~s}$ if the initial current is $20.0 \mathrm{~A}$ .

Question 16

The current to a capacitor is given by $i=6 t^{2}+t^{3}$ . If the initial change on the capacitor is $3.75 \mathrm{C}$ , find the charge at $2 \mathrm{~s}$ .

Question 17

The current to a capacitor is given by $i=t \sqrt{t^{2}+7.5}$ . The initial charge on the capacitor is 4.15 coulombs. Find an expression for the charge on the capacitor.

Question 18

A 13.5-F capacitor has an initial voltage of $75.0 \mathrm{~V}$ . It is charged with a current given by $i=t^{3 / 2}$ . Find the voltage across the capacitor at $2.50 \mathrm{~s}$ .

Question 19

The voltage across a $17.0-\mathrm{H}$ inductor is $v=\sqrt{2 t+32}$ . Find the current at $5.00 \mathrm{~s}$ if the initial current is $15.0 \mathrm{~A}$ .

Question 20

Find the equation for the charge in a capacitor if the current is $i=\frac{2 t}{t^{2}-1}$ and there is an initial charge of $25 \mathrm{C}$ .

Question 21

Find the equation for the current in a $25 \mathrm{H}$ inductor with a voltage of $v=\frac{4}{\sqrt{2 t+7}}$ and a current of $45 \mathrm{~A}$ at $t=0.75 \mathrm{~s}$ .

Question 22

Find the equation for the voltage across an $8.0 \mathrm{~F}$ capacitor with an initial voltage of $50.0 \mathrm{~V}$ if the current is $i=6 t^{2}-8 t+5$ .

Question 23

A 40-H inductor with a voltage of $v=\sqrt{t}+4$ and an initial current of $25 \mathrm{~A}$ . Find the current at $15 \mathrm{~s}$ .

Question 24

A 12-F capacitor with an initial voltage of $60 \mathrm{~V}$ . if the current is $i=15 t^{2}-6 t+7$ , find the voltage at $4 \mathrm{~s}$ .

Question 25

Find the area bounded by the curve $y=(x-3)^{2}-4$ and the $x$ -axis. Express your answer to four significant digits.

Question 26

Find the area bounded by $y=(x-2)^{2}$ and $y=8-x$ . Express your answer to four significant digits.

Question 27

The work done on an object, in joules, is equal to the area under the curve on the graph below of the force $F$ on the object vs the distance $x$ over which $F$ is applied to the object. For an object on the end of a spring with a force constant of $2 \mathrm{~N} / \mathrm{m}$ , the force of the spring on the object is given by $F=-2 x \mathrm{~N}$ . Find the work done by the spring when $x$ goes from $-3 \mathrm{~m}$ to $0 \mathrm{~m}$ .

Question 28

Find the area bounded by the curve $y=x^{2}-9$ and the line $y=2 x+6$ .

Question 29

Find the area bounded by the curve $y=\frac{1}{x}$ , the $y$ -axis, and the lines $x=2$ and $x=7$ . Express your answer to three significant digits,

Question 30

Find the area bounded by the parabolas $y=x^{2}+2 x-1$ and $y=-x^{2}+4 x+3$ .

Question 31

Find the area bounded by the curve $y=x^{2}-5 x$ and the line ${y=\frac{1}{2} x+8}$ . Round your answer to three significant digits.

Question 32

Find the area bounded by the curve $x=y^{2}$ and the line $x-y-6=0$ . Express your answer to three significant digits.

Question 33

Find the area between the curves $y=x^{2}-2$ and $y=\frac{1}{3} x+5$ . Round your answer to three significant digits.

Question 34

Find the area between the curves $y=2 \sqrt{x+4}$ and $x-4 y+4=0$

Question 35

Find the area between the curves $x=0.25 y^{2}-2$ and-. Round your answer to three significant digits.

Question 36

Find the area between the curves $x=y^{2}-9$ and $y=x$ . Round your answer to three significant digits.

Question 37

Find the area between the curves $y=x^{2}-4$ and $y=2 x-x^{2}$ .

Question 38

Find the volume generated by rotating the first quadrant area bounded by the set of curves around the $x$ -axis: $y=3 \sqrt{x}$ and $x=4$ . Express your answer to four significant digits.

Question 39

Find the volume generated by rotating the first quadrant area bounded by the curve around the $y$ -axis: $16 y^{2}+25 x^{2}=400$ . Express your answer to four significant digits.

Question 40

Find the volume generated by revolving the first quadrant area bounded by the set of curves around the indicated axis: $y=\sqrt{x}$ and $x=4$ , around $y=-2$ . Express your answer to three significant digits.

Question 41

Find the volume generated by rotating the first-quadrant area bounded by the curve $y=2 x^{3}$ , the line $x=2$ , and the $x$ -axis about the $x$ -axis. Express your answer to four significant digits.

Question 42

Find the volume generated by rotating the first-quadrant area bounded by the curve $y=2 x^{3}$ , the line $x=2$ , and the $x$ -axis about the $y$ -axis. Express your answer to three significant digits.

Question 43

Find the volume generated by rotating the area bounded by the curves $y=2 x^{2}$ and $y=x^{2}-2 x+3$ about the line $x=2$ . Express your answer to four significant digits.

Question 44

Find the volume generated by rotating the area bounded by $x=1$ and $x=y^{2}$ about $y=1$ . Express your answer to three significant digits.

Question 45

Find the volume generated by rotating the first quadrant area bounded by the curve $y=3 x^{2}-x^{3}$ around the $y$ -axis.

Question 46

Find the volume generated by rotating the first quadrant area bounded by the curves $x=y^{2}$ and $x=\sqrt[3]{y}$ around the $x$ -axis. Express your answer to three significant digits.

Question 47

Find the volume created by rotating the area between $y=\sqrt{3 x}$ , the x-axis and $x=12$ around the $x$ axis. Express your answer to four significant digits.

Question 48

The area bounded by the curve $x=-4 y^{2}+40 y-96$ and the $y$ -axis is rotated around the $y$ -axis. Find the volume. Express your answer to three significant digits.

Question 49

Find the volume created by rotating the first quadrant area between $x=2 \sqrt{y+1}$ , the $y$ -axis and $y=3$ around the y-axis. Express your answer to three significant digits.

Question 50

The area bounded by the curve $y=4-x^{2}$ and the x-axis is rotated around the x-axis. Find the volume to four significant digits.