I am confused on how to solve this. NOTE: the lower limit for the second function is (-6) NOT 6. Wouldn't let me type it in correctly.[tex]\int\limits^6_3 {|x-4|} \, dx + \int\limits^0_6 {\sqrt{36-x^2} } \, dx[/tex]
Answers
I think how you approach this depends on your knowledge of calculus.
If you don't know how to compute definite integrals yet, but you do know that they represent signed areas under curves, then you can plot both curves |x - 4| and √(36 - x ²), then recognize that the areas represented by these integrals are areas of geometric shapes. (See attached images)
First integral: if you plot |x - 4| on the interval [3, 6], you'll see that the integral corresponds to the area of two triangles. One of them has base = height = 1, and the other has base = height = 2. Then
[tex]\displaystyle\int_3^6|x-4|\,\mathrm dx=\frac12\times1\times1+\frac12\times2\times2=\frac52[/tex]
Second integral: if [tex]y=\sqrt{36-x^2}[/tex], then [tex]x^2+y^2=6^2[/tex], meaning this curve is the upper half of a circle with radius 6. On the interval [-6, 0], the area amounts to 1/4 of the total area of such a circle, so that
[tex]\displaystyle\int_{-6}^0\sqrt{36-x^2}\,\mathrm dx=\frac{\pi\times6^2}4=9\pi[/tex]
* * *
If you already know a few things about calculus and integration, you can compute these areas directly.
First integral:
[tex]\displaystyle\int_3^6|x-4|\,\mathrm dx=\int_3^4(4-x)\,\mathrm dx+\int_4^6(x-4)\,\mathrm dx[/tex]
[tex]\displaystyle\int_3^6|x-4|\,\mathrm dx=\left(4x-\frac{x^2}2\right)\bigg|_{x=3}^{x=4}+\left(\frac{x^2}2-4x\right)\bigg|_{x=4}^{x=6}[/tex]
[tex]\displaystyle\int_3^6|x-4|\,\mathrm dx=\left(\left(4\times4-\frac{4^2}2\right)-\left(4\times3-\frac{3^2}2\right)\right)+\left(\left(\frac{6^2}2-4\times6\right)-\left(\frac{4^2}2-4\times4\right)\right)[/tex]
[tex]\displaystyle\int_3^6|x-4|\,\mathrm dx=\frac52[/tex]
Second integral:
Substitute x = 6 sin(t ) and dx = 6 cos(t ) dt, then
[tex]\displaystyle\int_{-6}^0\sqrt{36-x^2}\,\mathrm dx=\int_{-\frac\pi2}^0\sqrt{6^2-(6\sin(t))^2} (6\cos(t))\,\mathrm dt[/tex]
[tex]\displaystyle\int_{-6}^0\sqrt{36-x^2}\,\mathrm dx=36\int_{-\frac\pi2}^0\cos (t) \sqrt{1-\sin^2(t)} \,\mathrm dt[/tex]
[tex]\displaystyle\int_{-6}^0\sqrt{36-x^2}\,\mathrm dx=36\int_{-\frac\pi2}^0\cos (t) \sqrt{\cos^2(t)} \,\mathrm dt[/tex]
[tex]\displaystyle\int_{-6}^0\sqrt{36-x^2}\,\mathrm dx=36\int_{-\frac\pi2}^0\cos (t) |\cos(t)| \,\mathrm dt[/tex]
For t ∈ [-π/2, 0], cos(t ) > 0, so |cos(t )| = cos(t ) :
[tex]\displaystyle\int_{-6}^0\sqrt{36-x^2}\,\mathrm dx=36\int_{-\frac\pi2}^0\cos^2(t)\,\mathrm dt[/tex]
Recall the half-angle identity,
cos²(t ) = (1 + cos(2t )) / 2
[tex]\displaystyle\int_{-6}^0\sqrt{36-x^2}\,\mathrm dx=18\int_{-\frac\pi2}^0(1+\cos(2t))\,\mathrm dt[/tex]
[tex]\displaystyle\int_{-6}^0\sqrt{36-x^2}\,\mathrm dx=18\left(t+\frac{\sin(2t)}2\right)\bigg|_{t=-\frac\pi2}^{t=0}[/tex]
[tex]\displaystyle\int_{-6}^0\sqrt{36-x^2}\,\mathrm dx=18\left(\left(0+\frac{\sin(2\times0)}2\right)-\left(-\frac\pi2+\frac{\sin\left(2\times\left(-\frac\pi2\right)\right)}2\right)\right)[/tex]
[tex]\displaystyle\int_{-6}^0\sqrt{36-x^2}\,\mathrm dx=9\pi[/tex]
Related Questions
help please<333 with an explanation <333 :]]]
Answers
the tax on a purchase of $20 is $2.80. How much tax will there be on a purchase of $45.50?
Answers
Answer:
$6.37
Step-by-step explanation:
You divide 2.80 by 20 to find the percentage of tax then you simply multiply your percentange by the cost. In this problem the tax was 14% or .14, so you multiply .14 by 45.50 to find the amount of tax on the purchase
ILL MARK BRAINLYEST
Which statement is true. Supplementary angles:
Group of answer choices
are congruent
add up to 180 degrees
are two angles inside a triangle
add up to 90 degrees
Answers
Step-by-step explanation:
Supplementary angles add up to 180 degrees
Complementary angles add up to 90 degrees
It is just a rule that is important to geometry
I hope that helps :)
Answer:
Add up to 180 degrees
Step-by-step explanation:
Supplementary angles can be two separate angles and the acute angles of a right triangle are complementary which make option 3 false. They are not always congruent and do not add up to 90 degree in total. Making option 1 and 4 false as well.
Jaleah needs some string to make bracelets. She needs
enough string to make 32 bracelets. Each bracelet takes 9 3/4
inches of string. How much string will Jaleah need to make
the bracelets?
Answers
Answer:
312
Step-by-step explanation:
3/4 × 32 = 24
9 × 32 = 288
288 +24=312
What ordered pairs name the locations of points P and Q?
Answers
Answer:
See below
Step-by-step explanation:
P(-4, 2)
Q(-3, 0)
PLEASE HELP!!!!!! ILL GIVE BRAINLIEST *EXTRA 40 POINTS** !! DONT SKIP :((
Answers
Can someone please help!!!
Answers
Answer: {t| t ≥ 2}
Step-by-step explanation:
3t - 6 ≥ t - 2
-t -t
---------------------
2t - 6 ≥ -2
+6 +6
---------------------
2t ≥ 4
t ≥ 2
A school bus carries 20 students. There are
currently 215 students in the 7th grade. How many
school buses are needed to take the 7th grade on a
field trip?
Answers
Answer:
11 buses
Step-by-step explanation:
215 divided by 20 is 10.75 and rounding up makes more sense for this problem, so 11 buses would be needed to carry the entire 7th grade.
Because of bad weather, the number of days next week that the captain of a charter fishing boat can leave port is uncertain. The following probability distribution for the number of days that the boat is able to leave port per week was determined based on historical data when the weather was poor: The number of days the boat can leave port per week Probability 0 0.20 1 0.05 2 0.05 3 0.05 4 0.15 5 0.10 6 0.15 7 0.25 Based on the probability distribution, what is the expected number of days per week the captain can leave port. Group of answer choices 3.55 4.00 3.50 4.05 2.85
Answers
Answer:
The expected number of days per week the captain can leave port is 4.05.
Step-by-step explanation:
We are given the following distribution:
[tex]P(X = 0) = 0.2[/tex]
[tex]P(X = 1) = 0.05[/tex]
[tex]P(X = 2) = 0.05[/tex]
[tex]P(X = 3) = 0.05[/tex]
[tex]P(X = 4) = 0.15[/tex]
[tex]P(X = 5) = 0.1[/tex]
[tex]P(X = 6) = 0.15[/tex]
[tex]P(X = 7) = 0.25[/tex]
What is the expected number of days per week the captain can leave port.
We multiply each outcome by its probability. So
[tex]E = 0*0.2 + 1*0.05 + 2*0.05 + 3*0.05 + 4*0.15 + 5*0.1 + 6*0.15 + 7*0.25 = 4.05[/tex]
The expected number of days per week the captain can leave port is 4.05.
Lainey brought a set of 22 markers for $6. What is the cost of 1 marker? Round answers to the nearest hundredths where necessary.
Answers
A right rectangular prism has a length of 6 centimeters, a width of 7 centimeters, and a height of 5 centimeters.
What is the volume of the prism?
____cm3
a) 214
b) 197
c) 210
CORRECT ANSWER= BRAINLY!!
Answers
Answer: c is the answer
Step-by-step explanation: the formuals for volume is length x width x height, 6x7x5, which is 210
Answer: C
Step-by-step explanation: Easy, just do length x width x height(l x w h). So, 6x7x5 = 210
Find the solution of the system of equations. -3x – 2y = 36 3х – 5y = 6
Answers
Answer:
y = -6
x = -8
Step-by-step explanation:
I hope you can read my handwriting.
Have a wonderful day!
Which equation has no real solutions?
) −2x2=−168
) 4x2+ 76 = 32
) 25x2= 1
) 43 −x2= 12
Answers
Answer:
Step-by-step explanation:
4x² + 76 = 32
4x² = -44
x² = -11
but x² must be non-negative, so there is no real solution.
Please help with this
Answers
Answer:
check pdf
Step-by-step explanation:
k over 6 plus 8 equals 5
Answers
Answer:
k= -18
Step-by-step explanation:
k/6 +8=5 (subtract 8 from both sides)
k/6= -3 (Multiply by 6 on both sides)
k=-18
The pet store had food priced at $10 and now is priced at $18. What is the percent markup?
Answers
Answer:
180%
Step-by-step explanation:
are z^3 and 3z equivalent expressions? explain.
Answers
Answer:
no, they are not.
Step-by-step explanation:
3z means a number multiplied by 3
so if z is 2, 3(2) = 6
but z^3 means z multiplied by z, and then multiplied by z again.
so if z is 2, it's 2x 2 = 4 x 2 = 8
let me know if you have questions
The two expressions are not equivalent to each other.
What are equivalent expressions?Equivalent expressions are expressions that are equal and synonyms to each other when used differently with various mathematical concepts.
From the expressions given:
z^33zThe first expression is a variable (z) raised to a power of 3. The power is also called an exponent.The second expression is a variable(z) that has a coefficient of 3Let us assume that:
z^3 = 3z
Now, let's replace 2 for the value of z on both sides, we have:
(2)^3 = 3(2)
8 ≠ 6
Learn more about equivalent expressions here:
https://brainly.com/question/2972832
Help me please i will give brainliest
Answers
2. C
3. D
Hopefully this is correct and helps!
Answer:
1. C. x^8
2. C. 3^9
3. D. 12x^7
Step-by-step explanation:
1.
x^6 * x^2 = x^(6 + 2) = x^8
Answer: C. x^8
2.
3^2 * 3^3 * 3^4 = 3^2 + 3 + 4) = 3^9
Answer: C. 3^9
3.
3x^2 * 4x^5 = 3 * 4 * x^2 * x^5 = 12x^(2 + 5) = 12x^7
Answer: D. 12x^7
ng 20-21
POSSIBLE POINTS: 10
In the school store, you can buy 20 calculators for $120. You want to buy 5 calculators. The table below represents the cost of different numbers of
calculators. Complete the table and explain the mathematical process you used to determine the cost of 5 calculators.
Answers
you would spend $30
Step-by-step explanation:
if 20 calculator cost 120 then you would divide 20 by 120 which would give you 6 dollars per calculator then you would multiply that by 5 which would give you a $30 for five calculators.
Answer:
where is the table?
I am just telling you how to get the cost of 5 calculators...
Step-by-step explanation:
Cost of 20 cal.=$120
Cost of 1 cal.=120/20
So,
cost of 5 calculators=
120/20*5
=$30
Hope it helps you
A triangle has sides with lengths of 4 meters, 5 meters, and 6 meters. Is it a right triangle?
Answers
Answer:
no
Step-by-step explanation:
because no pythagoras formula work here
Answer:
Lets solve it by phythagoras therom
[tex]alt^{2}[/tex] + [tex]base^{2}[/tex] = [tex]hypo^{2}[/tex]
[tex]6^{2}[/tex] = [tex]4^{2}[/tex] + [tex]5^{2}[/tex]
so, [tex]4^{2}[/tex] + [tex]5^{2}[/tex] would be equal to [tex]6^{2}[/tex]
[tex]4^{2}[/tex] + [tex]5^{2}[/tex] = 16 + 25
= 41
[tex]6^{2}[/tex] = 36
41 is not equal to 36
So it is not a right angled tringle.
Hope it helps
Please mark me as the brainliest
thank you
Helpppp!!! Pleaseeee
Answers
Answer:I’m pretty sure it’s 90
Step-by-step explanation:
Answer:
tan 60=√3/t
t=√3/√3=1 is your answer
A group of five students have a mean test score of "34.2." Another student has a score of 50. What is the mean average of all "six" students?
Answers
Answer:
The mean average of all "six" students is 36.83.
Step-by-step explanation:
Mean:
The mean is the sum of all scores divided by the number of scores.
A group of five students have a mean test score of "34.2." Another student has a score of 50.
This means that the sum is given by:
[tex]S = 5*34.2 + 50 = 171 + 50 = 221[/tex]
What is the mean average of all "six" students?
Sum of 6 students is 221. So
[tex]M = \frac{221}{6} = 36.83[/tex]
The mean average of all "six" students is 36.83.
9. The radius of sphere is increasing at a rate of 3cm/minute. Find the rate of change of volume
when r=2cm *
V = 12
48л
16л
64 л
32T
Answers
Answer:
48π
Step-by-step explanation:
Given that,
The radius of the sphere is increasing at a rate of 3cm/minute, [tex]\dfrac{dr}{dt}=3\ cm/minute[/tex]
The volume of a sphere is given by :
[tex]V=\dfrac{4}{3}\pi r^3[/tex]
Differentiating both sides wrt t.
[tex]\dfrac{dV}{dt}=\dfrac{4}{3}\pi \times 3r^2\times \dfrac{dr}{dt}\\\\=4\pi r^2\times \dfrac{dr}{dt}[/tex]
Put r = 2 cm and dr/dt = 3 cm/minute
So,
[tex]\dfrac{dV}{dt}=4\pi r^2\times \dfrac{dr}{dt}\\\\\dfrac{dV}{dt}=4\pi \times 2^2\times 3\\\\\dfrac{dV}{dt}=48\pi[/tex]
Hence, the correct option is (a) "48π".
Maria is starting a small DVD business online. She makes $3 on each DVD she sells. To start her business though, she had to invest $750. How many DVDs does she need to sell before she starts to make a profit?
Answers
Answer:
750 divided by 3. Equals 250
Step-by-step explanation:
If it says how much is needed and it gives the amount the person has, you will need to divide.
10
8
6
Distance (mi)
4
2.
o
20 30 40 50
Time (min)
The graph above shows Tomasso's rate
while jogging. Which of the following is an
ordered pair on the graph?
A (10, 1)
B (15, 3)
C (30,5)
Answers
Answer:
the answer should be A as thats what its showing on the graph
dy/dx=2xy/x²+y² solve
Answers
Answer:
[tex]y=\frac{-1}{2} \sqrt[]{4x^2+4c_{1}^2} -c_{1}\ or \ \frac{1}{2} \sqrt[]{4x^2+4c_{1}^2} -c_{1}[/tex]
Step-by-step explanation:
As it is first order nonlinear ordinary differential equation
Let y(x) = x v(x)
2xy/(x²+y²)=2v/(v^2+1)
dy=xdv+vdx
dy/dx=d(dv/dx)+v
x(dv/dx)+v=(2v)/(v^2+1)
dv/dx=[(2v)/(v^2+1)-v]/x
[tex]\frac{dv}{dx}=\frac{-(v^2-1)v}{x(v^2+1)}[/tex]
[tex]\frac{v^2+1}{(v^2-1)v}dv=\frac{-1}{x}dx[/tex]
∫[tex]\frac{v^2+1}{(v^2-1)v}dv[/tex] = ∫[tex]\frac{-1}{x}dx[/tex]u=v^2
du=2vdv
Left hand side:
∫[tex]\frac{v^2+1}{v(v^2-1)}dv[/tex]
=∫[tex]\frac{u+1}{2u(u-1)}du[/tex]
=[tex]\frac{1}{2}[/tex]∫[tex](\frac{2}{u-1} -\frac{1}{u} )du[/tex]
=[tex]ln(u-1)-\frac{ln(u)}{2} +c[/tex]
=[tex]ln(v^2-1)-\frac{ln(v^2)}{2}+c[/tex]
Right hand side:
[tex]=-ln(x)[/tex]
Solve for v:
[tex]v=-\frac{-c_{1}+\sqrt{c_{2}+4x^2 } }{2x} \ or \ \frac{-c_{1}+\sqrt{c_{2}+4x^2 } }{2x}\\[/tex]
[tex]y=\frac{-1}{2} \sqrt[]{4x^2+4c_{1}^2} -c_{1}\ or \ \frac{1}{2} \sqrt[]{4x^2+4c_{1}^2} -c_{1}[/tex]
20 points!!!!! The table represents a linear function
Answers
If F(x) = 5x - 7 and G(X) = x2 - 41, what is G(F(x))?
O A. 5(x2 - 41) -7
O B. -5x +7x2 + 205x – 287
O C. (5-7 - 41
O D. 5x2 - 7x2 - 205x+287
Answers
Which of the following shapes is a prism? Select All that apply.
A. Shape 1: Circle
B. Shape 2: Cylinder
CShape 3: Rectangular Box
D. Shape 4: Cube
E. Shape 5: Pyramid
F. Shape 6: Cone
Answers
Answer:
C D
Step-by-step explanation:
can somebody help me?
Answers
Answer:
15x + 7
Step-by-step explanation:
i set this up
8x + 16 + 7x -9
the next step is to combine like terms
8x + 7x = 15x
16 - 9 = 7
15x = 7
15x + 7