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
The angle measurements in the diagram are represented by the following expressions
Answers
Answer:
∠A is equal to 73°
Step-by-step explanation:
Because the two lines that move roughly bottom/left to top/right are marked as being parallel, we can say that ∠A + ∠B = 180°
With that we can take the two equations at the top, and add them together for a sum of 180°:
6x - 35° + 3x + 53° = 180°
9x + 18° = 180°
x + 2° = 20°
x = 18°
Now that we have x, we can solve for ∠A:
∠A = 6x - 35°
∠A = 6(18°) - 35°
∠A = 108° - 35°
∠A = 73°
We can test our answer by solving for ∠B. We know that ∠A and ∠B add up to 180°, so ∠B = 180° - ∠A = (180 - 73)° = 107°. Now let's solve for be given x and the original equation, and see if we have the right numbers:
∠B = 3x + 53°
∠B = 3(18°) + 53°
∠B = 54° + 53°
∠B = 107°
Which matches up, showing us that we have the correct answer.
3/4 -5/8 in simplest form plz
Answers
Hello! May someone please help on this :)
I would probably give brainliest.
Answers
Answer:
theres 16 doughnuts
Step-by-step explanation:
geometry proofs help please
Answers
Answer:
trust your instincts
Step-by-step explanation:
I can not interfere with your destiny you will need to decide your destiny
The director of a customer service center wants to estimate the mean number of customer calls the center handles each day, so he randomly samples 29 different days and records the number of calls each day. The sample yields a mean of 279.4 calls with a standard deviation of 25.1 calls per day. He can be 99% confident that the mean number of calls per day is between 266.5 and ________. (Round your answer to 1 decimal place.)
Answers
Answer:
He can be 99% confident that the mean number of calls per day is between 266.5 and 292.3.
Step-by-step explanation:
Confidence interval:
A confidence interval is symmetric, which means that the difference between the sample mean and the lower bound is the same as the difference between the upper bound and the sample mean.
In this question:
Sample mean: 279.4 calls
Lower bound: 266.5 calls
Upper bound: x calls
Due to the symmetry of the confidence interval:
[tex]x - 279.4 = 279.4 - 266.5[/tex]
[tex]x = 12.9 + 279.4[/tex]
[tex]x = 292.3[/tex]
So
He can be 99% confident that the mean number of calls per day is between 266.5 and 292.3.
3. Three model greenhouses are shown below. They are made by covering the tops and sides of a wood frame with plastic. The bottom of the green house is not covered with plastic. Of the three models, which has the greatest amount of plastic needed to make it and which has the least amount of plastic needed? 3 ft ) 5 ft 5 ft 5 ft 3111/ 8 ft 6 ft 6 it 8 ft 8 ft 6 ft 8 ft Model A 12 ft Model B 10 ft Model C
Answers
Answer:
New Materials Have Made Greenhouses Less ... In our garden, we recently had a major ... Besides the trends mentioned above, there are many ... Since seasonal rains will not reach your greenhouse plants, you need to ... Glass and clear plastic function in basically the ...
Step-by-step explanation:
Thinking about the table...Why does this
table start at zero? Why aren't there any
negative numbers in the DOMAIN?
Answers
Answer:
Why does this table start at zero?The vertical axis on trending charts should almost always start at zero. The reason it's almost always is because you may have trending data that contains negative values or fractions. In those situations, it's generally best to keep Excel's default scaling.
Why aren't there any negative numbers in the DOMAIN?There are no domain restrictions, even though there is a variable under the radical. Since , can never be negative. The least it can be is , so there is no danger of taking the square root of a negative number.
Step-by-step explanation:
Hope it is helpful....
Help ASAP!!!! Thanks
Answers
Answer:
No DB is not a perpendicular bisector of AC
Step-by-step explanation:
This is because as AC is a straight line it's angle degree is 180 which when bisected by DB becomes,
180 ÷ 2 = 90
On both the angles i.e <BDC = <NDA = 90°
To make it a perpendicular bisector but
<BDC is not equal to <NDA is not equal to 90°.
Hence, DB is not a perpendicular bisector of line AC.
Find the total surface area of the
following prism:
5 cm
5 cm
5 cm
5 cm
10 cm
5 cm
apothem = 3.6 cm
SA = [ ? ] cm2
Answers
Answer:
Total surface area = 340 cm²
Step-by-step explanation:
Surface area of the given prism = Area of two bases + Area of its lateral surfaces
Area of pentagonal base = [tex]\frac{1}{2}(\text{Apothem)}(\text{Perimeter})[/tex]
= [tex]\frac{1}{2}(3.6)(5\times 5)[/tex]
= 45 cm²
Area of the lateral sides of the prism = Perimeter of the pentagonal base × height
= (5×5)×10
= 250 cm²
Total surface area of the prism = 2(45) + 250
= 90 + 250
= 340 cm²
the length of a rectangle is represented by 3x-7 and the width by x+4. express the perimeter as a polynomial in simplest form
Answers
Answer: 8x - 6
Step-by-step explanation:
Perimeter is found by doing 2l + 2w
The length is 3x - 7 and width is x + 4
2(3x - 7) + 2(x + 4) =
6x - 14 + 2x + 8 =
8x - 6
Help please. Assignment due tonight.
Answers
Answer:
sin b
Step-by-step explanation:
Answer:
cos A
Step-by-step explanation:
sine = [tex]\frac{opposite}{hypotenuse}[/tex]
sine B = [tex]\frac{3}{5}[/tex]
cos = [tex]\frac{adjacent}{hypotenuse}[/tex]
cos A = [tex]\frac{3}{5}[/tex]
A boat traveled one hundred fifty feet in 9.7 seconds. How far would the boat travel in one minute thirty seconds?
Answers
Answer:
40250
Step-by-step explanation:
Raina drinks 0.65 L of milk every day. How much milk does she drink in 9 days?
Write your answer in milliliters.
Answers
Answer: 5850 milliliters
Step-by-step explanation:
First, you have to do 0.65L x 9 days = 5.85L
And then you have to convert it to milliliters so:
5.85L= 5850 milliliters
Describe the slope of the line.
The slope is
Find the slope.
m =
step by step explanation
Answers
Answer:
The slope is POSITIVE.
The line has a slope of positive 4/3
Step-by-step explanation:
It rises 4 then runs 3.
How do I answer this ? (1 + 3x)º
44°
Answers
Answer:
132x+44
Step-by-step explanation:
(1+3x)(44)
=(1+3x)(44)
=(1)(44)+(3x)(44)
=44+132x
=132x+44
I need help on this part
Answers
Answer:
ang 4 = 60°
Step-by-step explanation:
ang 1 and ang 4 are Linear pair or supplementary angles.
ang 1 + ang 4 = 180
120° + ang 4 = 180° (Putting the value of ang 1)
ang 4 = 180° - 120°
ang 4 = 60°
Subtract the fractions. Simplify if possible. 3/4 - 1/4 - 2/4 =
Answers
Answer is 0
Reflect the figure across the y- axis, then state the ordered pairs of the image.
May someone please explain how I would reflect it?
Answers
R’ (2, -3)
X’ (-2, 0)
Z’ (-2, -2)
EXPLANATION:
Reflecting a figure across the y-axis basically means flipping all of the x-values of your coordinates while retaining its y-value.
In other words, flip the signs of x and keep y the exact same for each coordinate.
B (-3, 2)
=
B’ (3, 2)
I just need the right answer with steps
Answers
y - 6 + 6 = 2x + 10 + 6
y = 2x + 16
So y = 2x + 16 is your answer
Nikki brought a patio set on sale for $480 the original price was 850 what is the rate of discount
Answers
Answer:
43.5%
Step-by-step explanation:
Substitute the original price of $850 and the sale price of $480 to find
amount of discount=850−480=370
Use the amount of discount, $370, to find the discount rate. Use the equation
amount of discount=discount rate×original price.
Rearrange the equation to solve for the unknown value.
discount rate=amount of discountoriginal price
Substitute in the known values and solve to find the discount rate.
370850=0.435=43.5%
Thus, the discount rate on the patio set was 43.5%.
I need help with this question !!
Answers
Answer:
Step-by-step explanation:
Yes, A-A-A
find the number of terms in following algebraic expressions.
Answers
Answer:
it needs time wait for 10 min.
You are making a poster to support your friend for homecoming. Your original photo is 4 inches by 6 inches. You want the photo on the poster to be 2 feet by 3 feet. What is the scale factor of this dilation? show work pls :(
Answers
Answer:
6
Step-by-step explanation:
2 ft = 24in so 4 to 24 is times 6
3ft=36 in and 6 to 46 is times 6
Answer:
6/1
Step-by-step explanation:
2ft=24 inches
3ft=36 inches
poster/original photo
24in/4in=6/1
36in/6in=6/1
6/1
whoever answers this correctly gets brainliest
Answers
Answer:
Democratic is the one that says the government should play an active role. Republican is the second choice (should be limited)
Answer: For the first, (goverment should play an active role in resolving social or economic issues) would be the Democratic Party.
And the Second (Government power should be limited and not used to resolve social or economic issues) is the republican.
A B C D ..........zzz
Answers
Answer:
QA = DU
yakno with the squiggly line over the equals sign
A pizza dinner for $20 was marked down 20%. What was the new sale cost of the dinner?
Answers
Answer:
$16
Step-by-step explanation:
take 20 multiply it by 0.2 and you get 4 multiplied by 4 is 16 so it is 16 dollars
the sumof the present ages of a father and his son is 60 years. 6years ago, the father's age was 5 times the age of his son.after 6 years, how old will the son be?
Answers
Answer:
20 years
Step-by-step explanation:
Let the present ages of son and father be x and (60 -x) years respectively.
Then, (60 - x) - 6 = 5(x - 6)
54 - x = 5x - 30
6x = 84
x = 14.
Son's age after 6 years = (x+ 6) = 20 years..
I hope this helps a little bit.
How many cubic feet will it take to the
prism below?
A. 560 ft cubed
B. 480 ft cubed
C. 280 ft cubed
D. 210 ft cubed
Answers
Answer:
C. 280 ft cubed
Step-by-step explanation:
1/2x7x5x16=280
determine the 1st term of the first 21 term of an arithmetic series if the sum of the three middle term is 126 and the sum of the three last terms is 288
Answers
Answer:
- 18
Step-by-step explanation:
arithmetic sequences are in form:
a(n)=a(o)+d(n-1)
sum are in form:
s(n)=[a(0)+a(n)](n/2)
three middle terms is:
10th, 11th, 12th term
How to think: 21 = 10+1+10 --> middle 3 is 10,11,12
sum of these 3 terms:
a(10)+a(11)+a(12)
=a(o)+9d+a(0)+10d+a(0)+11d
=3a(0)+30d=126
a(0)+10d=42 ......(equation 1)
Last three terms is:
19th, 20th, 21th
a(19)+a(20)+a(21)
=a(0)+18d+a(0)+19d+a(0)+20d
=3a(0)+57d=288
a(0)+19d=96 ......(equation 2)
now you have
a(0)+10d=42 ......(equation 1)
a(0)+19d=96 ......(equation 2)
solve and gives
d=6
a(0)= -18
Please explain in words.
Answers
Answer:
No
Step-by-step explanation:
When you are adding a positive by a negative, simply subtract and take the sign of the larger number.