# 5.4 Integration formulas and the net change theorem  (Page 5/8)

 Page 5 / 8

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.

$12\pi {R}^{2}=8\pi {\int }_{R}^{2R}rdr$

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.

Suppose that a particle moves along a straight line with velocity $v\left(t\right)=4-2t,$ where $0\le t\le 2$ (in meters per second). Find the displacement at time t and the total distance traveled up to $t=2.$

$d\left(t\right)={\int }_{0}^{t}v\left(s\right)ds=4t-{t}^{2}.$ The total distance is $d\left(2\right)=4\phantom{\rule{0.2em}{0ex}}\text{m}\text{.}$

Suppose that a particle moves along a straight line with velocity defined by $v\left(t\right)={t}^{2}-3t-18,$ where $0\le t\le 6$ (in meters per second). Find the displacement at time t and the total distance traveled up to $t=6.$

Suppose that a particle moves along a straight line with velocity defined by $v\left(t\right)=|2t-6|,$ where $0\le t\le 6$ (in meters per second). Find the displacement at time t and the total distance traveled up to $t=6.$

$d\left(t\right)={\int }_{0}^{t}v\left(s\right)ds.$ For $t<3,d\left(t\right)={\int }_{0}^{t}\left(6-2t\right)dt=6t-{t}^{2}.$ For $t>3,d\left(t\right)=d\left(3\right)+{\int }_{3}^{t}\left(2t-6\right)dt=9+\left({t}^{2}-6t\right).$ The total distance is $d\left(6\right)=9\phantom{\rule{0.2em}{0ex}}\text{m}\text{.}$

Suppose that a particle moves along a straight line with acceleration defined by $a\left(t\right)=t-3,$ where $0\le t\le 6$ (in meters per second). Find the velocity and displacement at time t and the total distance traveled up to $t=6$ if $v\left(0\right)=3$ and $d\left(0\right)=0.$

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 $v\left(t\right)$ and the height $h\left(t\right)$ of the ball t seconds after it is thrown and before it returns to the ground.

$v\left(t\right)=40-9.8t;h\left(t\right)=1.5+40t-4.9{t}^{2}$ m/s

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 $v\left(t\right)$ and the height $h\left(t\right)$ of the ball t seconds after it is thrown and before it returns to the ground.

The area $A\left(t\right)$ 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.

The net increase is 1 unit.

A spherical balloon is being inflated at a constant rate. If the volume of the balloon changes from 36 π in. 3 to 288 π in. 3 between time $t=30$ and $t=60$ seconds, find the net change in the radius of the balloon during that time.

Water flows into a conical tank with cross-sectional area πx 2 at height x and volume $\frac{\pi {x}^{3}}{3}$ up to height x . If water flows into the tank at a rate of 1 m 3 /min, find the height of water in the tank after 5 min. Find the change in height between 5 min and 10 min.

At $t=5,$ the height of water is $x={\left(\frac{15}{\pi }\right)}^{1\text{/}3}\phantom{\rule{0.2em}{0ex}}\text{m}\text{.}.$ The net change in height from $t=5$ to $t=10$ is ${\left(\frac{30}{\pi }\right)}^{1\text{/}3}-{\left(\frac{15}{\pi }\right)}^{1\text{/}3}$ m.

A horizontal cylindrical tank has cross-sectional area $A\left(x\right)=4\left(6x-{x}^{2}\right){m}^{2}$ at height x meters above the bottom when $x\le 3.$

1. The volume V between heights a and b is ${\int }_{a}^{b}A\left(x\right)dx.$ Find the volume at heights between 2 m and 3 m.
2. Suppose that oil is being pumped into the tank at a rate of 50 L/min. Using the chain rule, $\frac{dx}{dt}=\frac{dx}{dV}\phantom{\rule{0.2em}{0ex}}\frac{dV}{dt},$ at how many meters per minute is the height of oil in the tank changing, expressed in terms of x , when the height is at x meters?
3. How long does it take to fill the tank to 3 m starting from a fill level of 2 m?

#### Questions & Answers

f(x) = x-2 g(x) = 3x + 5 fog(x)? f(x)/g(x)
fog(x)= f(g(x)) = x-2 = 3x+5-2 = 3x+3 f(x)/g(x)= x-2/3x+5
diron
pweding paturo nsa calculus?
jimmy
how to use fundamental theorem to solve exponential
find the bounded area of the parabola y^2=4x and y=16x
what is absolute value means?
Chicken nuggets
Hugh
🐔
MM
🐔🦃 nuggets
MM
(mathematics) For a complex number a+bi, the principal square root of the sum of the squares of its real and imaginary parts, √a2+b2 . Denoted by | |. The absolute value |x| of a real number x is √x2 , which is equal to x if x is non-negative, and −x if x is negative.
Ismael
find integration of loge x
find the volume of a solid about the y-axis, x=0, x=1, y=0, y=7+x^3
how does this work
Can calculus give the answers as same as other methods give in basic classes while solving the numericals?
log tan (x/4+x/2)
Rohan
Rohan
y=(x^2 + 3x).(eipix)
Claudia
is this a answer
Ismael
A Function F(X)=Sinx+cosx is odd or even?
neither
David
Neither
Lovuyiso
f(x)=1/1+x^2 |=[-3,1]
apa itu?
fauzi
determine the area of the region enclosed by x²+y=1,2x-y+4=0
Hi
MP
Hi too
Vic
hello please anyone with calculus PDF should share
Which kind of pdf do you want bro?
Aftab
hi
Abdul
can I get calculus in pdf
Abdul
explain for me
Usman
okay I have such documents
Fitzgerald
Hamza
How to use it to slove fraction
Hello please can someone tell me the meaning of this group all about, yes I know is calculus group but yet nothing is showing up
Shodipo
You have downloaded the aplication Calculus Volume 1, tackling about lessons for (mostly) college freshmen, Calculus 1: Differential, and this group I think aims to let concerns and questions from students who want to clarify something about the subject. Well, this is what I guess so.
Jean
Im not in college but this will still help
nothing
how en where can u apply it
Migos
how can we scatch a parabola graph
Ok
Endalkachew
how can I solve differentiation?
with the help of different formulas and Rules. we use formulas according to given condition or according to questions
CALCULUS
For example any questions...
CALCULUS
v=(x,y) وu=(x,y ) ∂u/∂x* ∂x/∂u +∂v/∂x*∂x/∂v=1
log tan (x/4+x/2)
Rohan
what is the procedures in solving number 1?