30-32 sketch the region enclosed by the given curves and find its area.
31. [tex]y=\cos ^{2} x \sin x, \quad y=\sin x, \quad 0 \leqslant x \leqslant \pi[/tex]
Answers
Check the picture below.
so hmmm for the sake of completion, let's check where they intersect
[tex]cos^2(x)sin(x)=sin(x)\implies cos^2(x)sin(x)-sin(x)=0 \\\\\\ sin(x)[cos^2(x)-1]=0\implies \begin{cases} sin(x)=0\\ cos^2(x)-1=0 \end{cases} \\\\[-0.35em] ~\dotfill\\\\ sin(x)=0\implies x=sin^{-1}(0)\implies \boxed{x=0} \\\\[-0.35em] ~\dotfill\\\\ cos^2(x)-1=0\implies cos^2(x)=1\implies cos(x)=\sqrt{1} \\\\\\ cos(x)=1\implies x=cos^{-1}(1)\implies \boxed{x=\pi}[/tex]
now, notice, in the picture the function that is "above" or the "ceiling" function is the sin(x), so we'll get the area under the curve by using "above" - "below" functions.
[tex]\stackrel{above}{sin(x)}~~ - ~~\stackrel{below}{cos^2(x)sin(x)} \implies sin(x)-[1-sin^2(x)]sin(x) \\\\\\ sin(x)-[sin(x)-sin^3(x)]\implies ~~\begin{matrix} sin(x)-sin(x) \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~ +sin^3(x)\implies sin^3(x)[/tex]
now let's use the triple angle identity of sine
[tex]\stackrel{\textit{triple angle identity}}{sin(3x)=3sin(x)-4sin^3(x)}\implies 4sin^3(x)=3sin(x)-sin(3x) \\\\\\ sin^3(x)=\cfrac{3sin(x)-sin(3x)}{4}\implies sin^3(x)=\cfrac{3}{4}sin(x)-\cfrac{1}{4}sin(3x) \\\\[-0.35em] ~\dotfill\\\\ \displaystyle\int\limits_{0}^{\pi }~sin^3(x)dx\implies \int\limits_{0}^{\pi }~\left[ \cfrac{3}{4}sin(x)-\cfrac{1}{4}sin(3x) \right]dx[/tex]
[tex]\displaystyle \cfrac{3}{4}\int\limits_{0}^{\pi }sin(x)dx-\cfrac{1}{4}\int\limits_{0}^{\pi }sin(3x)dx\implies \left. \cfrac{3}{4} \cdot -cos(x) \right]_{0}^{\pi }-\left. \cfrac{1}{4} \cdot \cfrac{-cos(3x)}{3} \right]_{0}^{\pi } \\\\\\ \cfrac{3}{2}~~ - ~~\cfrac{1}{6}\implies \implies \blacktriangleright \cfrac{4}{3} \blacktriangleleft[/tex]
![30-32 Sketch The Region Enclosed By The Given Curves And Find Its Area. 31. [tex]y=\cos ^{2} X \sin](https://brainimage.s3.us-east-005.backblazeb2.com/answers/attachments/FtoPaefYJN1O4IqmHBEvSKTw4ZELIpUs.png)
Related Questions
PLEASE HELP!!!!
y = 1.4 (1.02)
a =
b=
r(as a percentage)
Answers
Answer:
a = 1.4 then b = (1+0.02) and r = 2%
There are 2022 kangaroos and some koalas living across seven parks. in each park the number of kangaroos is equal to the number of koalas in all the other parks. how many koalas live in the seven parks in total
Answers
Answer:
337
Step-by-step explanation:
I remember this from UKMT maths challenge.
Let the number of kangaroos in each of the seven parks be , , , , , and with the
corresponding number of koalas being , , , , , and . The question tells us that the
number of kangaroos in any park is equal to the sum of the numbers of koalas in the other six
parks. Therefore we have
= + + + + + ,
= + + + + + ,
= + + + + + ,
= + + + + + ,
= + + + + + ,
= + + + + + ,
= + + + + + .
Adding these equations, we obtain
+ + + + + + = 6( + + + + + + ).
The total number of kangaroos in the seven parks is 2022. Hence
+ + + + + + = 2022.
Therefore
2022 = 6( + + + + + + )
and hence the total number of koalas in the seven parks is
+ + + + + + = 2022 ÷ 6 = 337.
Someone please helppp
your friend house is 6.4 miles north and 12 miles west of your bank. how far is your friends house from your bank
Answers
Answer:
the answer is 76.8
Step-by-step explanation:
multiple of the number
Cavalieri’s principle states that if two solids of equal height have equal cross-sectional areas at every level parallel to the respective bases, then the two solids have equal volume. the two shaded solids both have a height of 2r units. at every level, the areas of the cross sections of both solids equal π(r2 – b2). a cylinder and a sphere are shown. 2 cones are cut out of the cylinder. the cones have a radius of 4. the cylinder has a height of 2 r. the sphere has a radius of 4. the sphere has a height of 2 r. cross-sectional areas are shown on each figure. therefore, the formula for the volume of the sphere can be derived by writing an expression that represents the volume of one cone within the cylinder. the two cones within the cylinder. the solid between the two cones and the cylinder. the cylinder.
Answers
Therefore, the formula for the volume of the sphere can be derived by writing an expression that represents the volume of C. The solid between the two cones and the cylinder
What is the Volume of a Sphere?
This refers to the capacity of a sphere, which is the maximum that can be contained in the sphere. This is denoted as V=4/3πr3
Hence, we can see that based on Cavalieri’s principle that states that if two solids of equal height have equal cross-sectional areas, then the two solids have equal volume.
Therefore, the two shaded solids both have a height of 2r units, and thus, to derive the function, we would have to write an expression that represents the volume of the solids between the two cones and the cylinder
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Answer: B
Step-by-step explanation: just did it
If you live in a state that needs to collect $18 million in property taxes and has a tax rate of 2.77%, what is that state's
taxable value?
O $650 million
O $65 million
O $6 million
O $499,000
Answers
Answer:
la respuesta es 65 millones
evauleate 150-(2+x)^3 when x is 2
Answers
Answer:
Down below
Step-by-step explanation:
150 - (2 + x)^3
150 - (2 + 2)^3
150 - (4)^3
150 - 64
= 86
I would check my math but I believe this is correct.
Answer:
86
150-(2+x)^3
⇒ x = 2
2 + 2 = 4
4^3(4 • 4 • 4) = 64
150 - 64 = 86
The financial planner for a beauty products manufacturer develops the system of equations below to determine how many combs must be sold to generate a profit. the linear equation models the income, in dollars, from selling x plastic combs; the quadratic equation models the cost, in dollars, to produce x plastic combs. according to the model, for what price is each comb being sold? startlayout enlarged left-brace 1st row y = startfraction x over 2 endfraction 2nd row y = negative 0.03 (x minus 95) squared 550 $0.03 $0.50 $0.95 $2.00
Answers
The price of each comb is $0.5, and the total income from selling x plastic combs is y = x/2
What is an equation?An equation is an expression that shows the relationship between two or more number and variables.
The linear equation y = x/2 models the income, in dollars, from selling x plastic combs. Hence:
Price of each comb = (x/2) / x = 0.5
The price of each comb is $0.5, and the total income from selling x plastic combs is y = x/2
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Answer: B. $0.50
Step-by-step explanation:
Took the Test/Exam
4.
Suppose events A and B are independent and P(A and B) is 0.2
Which statement could be true?
(1) P(A) = 0.4, P(B) = 0.3, P(A or B) = 0.5
(2) P(A) = 0.8, P(B) = 0.25
(3) P(A/B) = 0.2, P(B) = 0.2
(4) P(A) = 0.15, P(B) = 0.05
Answers
Answer: answer choice 2) P(A) = 0.8, P(B) = 0.25
Step-by-step explanation:
The True statement is P(A) = 0.8, P(B) = 0.25.
The correct option is (B).
What is Conditional Probability?Conditional probability is a measure of the probability of an event occurring, given that another event has already occurred.
1. 0.4 ·0.3≠ 0.2,
2. 0.8 · 0.25 = 0.2
3. P(A|B)= P(A)= 0.2
0.2 ≠ 0.2 · 0.2
4. 0.2 ≠ 0.15 · 0.05
Hence. the only condition is satisfied is (B).
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How do I find the domain range and function of the graph
Answers
Answer:
D = {-3, -2, -1, 1, 4}
R = {-5, -1, 0, 2}
Step-by-step explanation:
You have correctly described the process of finding the domain and range by the way you filled in the blanks at the top of the sheet.
__
The domain is the set of x-values of the points on the graph:
D = {-3, -2, -1, 1, 4}
The range is the set of (unique) y-values of the points on the graph:
R = {-5, -1, 0, 2}
The density of iron is 7.8 g/cm3. an iron beam has a volume of 10,504 cm³. what is the mass of the iron beam? enter your answer as a decimal in the box.
Answers
Under the assumption of uniform iron beam and definition of uniform density, the total mass of the iron beam is equal to 81931.2 grams.
How to determine the mass of the iron beam
Let suppose that the iron beam has an uniform density, the mass of the iron beam (m), in gram, is equal to the product of its density (ρ), in grams per cubic centimeter, and its volume (V), in cubic centimeters:
m = ρ · V (1)
If we know that ρ = 10504 cm³ and V = 7.8 g/cm³, then the mass of the iron beam is:
m = (7.8 g/cm³) · (10504 cm³)
m = 81931.2 g
Under the assumption of uniform iron beam and definition of uniform density, the total mass of the iron beam is equal to 81931.2 grams.
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The sum of three consecutive intergers is 189 what are the integers
Answers
Answer:
62, 63, 64
Step-by-step explanation:
The middle integer of three consecutive integers will be the average of them. That is, it will be their sum, divided by 3.
__
The middle integer is 189/3 = 63.
The three consecutive integers are 62, 63, 64.
_____
Additional comment
If you'd like to write an equation that tells you the same thing, let x represent the middle integer. Then the lower one is x-1, and the upper one is x+1. Their sum is ...
(x-1) +x +(x+1) = 189
3x = 189 . . . . . . . . . . . . simplify
x = 189/3 = 63 . . . . . . . divide by the coefficient of x
(x-1) = 63-1 = 62 . . . . . . find the other two integers
(x+1) = 63+1 = 64
What is the remainder of (x3+5x2−32x−7)÷(x−4) ? Enter your answer in the box.
Answers
Answer:
x^2+5x-36-7/x
Step-by-step explanation:
assuming x^3 and 5x^2
please solve this problem using the pythagorean theorem, and give me the right answer and I will give you brainliest. :)
Answers
Answer:
6.5
Step-by-step explanation:
9.1^2-6.4^2 is 41.85
square root of that is 6.469...
the food bill for meal at a restaurant is $12. THe sales tax is 6.5%, and you leave 15% tip. Wat is the total cost of the meal? PLEASE ANSWER FAST
Answers
Answer:
Step-by-step explanation:
12*6.5%=0.72
12+15%=13.8
0.72+13.8=14.52
What is the mean absolute deviation of 1, 2, 2, 3, 5, 6, 6, 8, 8, 9
Answers
Answer:
5
Step-by-step explanation:
1+2+2+3+5+6+6+8+8+9/10=5
what is the area in square centimeters of the isosceles trapezoid below?
Answers
65.28 cm^2
[tex]\bold{SOLUTION:}[/tex]
Find (qor)(x).
q(x) = 3x
r(x) = 6x + 5
Write your answer as a polynomial in simplest form.
Answers
Answer: 18x + 15
Step-by-step explanation:
(qor)(x)
q(r(x))
q(6x+5)
3(6x+5)
⇒18x + 15 This is your answer I believe
if we were solving for x then..
⇒x = [tex]-\frac{15}{18}[/tex]
If the dimensions of a rectangular prism are 10 cm x 5 cm x 7 cm, what is
the surface area?
Answers
Answer:
310
Step-by-step explanation:
Multiply the l/w/h x 2
The cosine of an angle a is 0.8. what is the value of sin (a)
Answers
Sine is opposite side over hypotenuse
You need 2 values to solve this.
A prism has a volume of 125 cubic inches. The base measures 5 in x 5 in What is the height? a. 10 inches b. 25 inches c. 5 inches d. 125 inches
Answers
Answer:C
Step-by-step explanation:First multiply 5*5=25 then divide 125/5 = 5
What is the volume of a sphere with a radius of 6 mm? sphere v = four-thirds pi r cubed v = 216pi mm3 v = 288pi mm3 v = 512pi mm3 v = 648pi mm3
Answers
Answer:
[tex]V =288 \pi \ mm^3[/tex]
Step-by-step explanation:
[tex]V =\frac{4}{3} \pi r^{3}[/tex]
[tex]=\frac{4}{3} \pi \left( 6\right)^{3}[/tex]
[tex]=\left( \frac{4}{3} \right) \times \pi \times 216[/tex]
[tex]=\left( \frac{4}{3} \right) \times 216\times \pi[/tex]
[tex]=288\times \pi[/tex]
Answer:
IT IS B
Step-by-step explanation:
SOURCE: TRUST ME BRO
Rita has a circular hot tub. the hot tub has a circumference of 25.12 feet. it is 3.5 feet deep.
how much water can the hot tub hold?
Answers
Volume is a three-dimensional scalar quantity. The amount of water that the tub can hold is 4,981.75 litres.
What is volume?A volume is a scalar number that expresses the amount of three-dimensional space enclosed by a closed surface.
Given the circumference of the tub is 25.12 feet, therefore, the radius of the circumference of the tub will be,
Circumference = 2πr
25.12 = 2×π×r
r = 3.9979ft ≈ 4ft
Now, the volume of the tub will be,
Volume = πr²h = π×(4²)×3.5 = 175.929 ft³
Since 1 ft³ is equal to 28.3168 litres, therefore, the amount of water that a tub can store is
Volume of water = 175 × 28.3168 = 4,981.75 litres
Hence, the amount of water that the tub can hold is 4,981.75 litres.
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Solve the following...
[tex]1 \cfrac{1}{5} \times 2\cfrac{1}{3} [/tex]
Answers
Let's see
[tex]\\ \rm\Rrightarrow 1\dfrac{1}{5}\times2\dfrac{1}{3}[/tex]
[tex]\\ \rm\Rrightarrow \dfrac{6}{5}\times \dfrac{7}{3}[/tex]
[tex]\\ \rm\Rrightarrow \dfrac{6(7)}{5(3)}[/tex]
[tex]\\ \rm\Rrightarrow \dfrac{42}{15}[/tex]
[tex]\\ \rm\Rrightarrow2 \dfrac{12}{15}[/tex]
Double tap to add title • AC is the diameter of the circle. Angle AWB is 120 degrees. How big is arc BC? A 1200 B w c D 85
Answers
Answer:
Step-by-step explanation:
What is the area,in square meters,of the isosceles trapezoid?
Answers
Step-by-step explanation:
the area of a trapezoid is
(base1 + base2)/2 × h = (18.6 + 9.2)/2 × 7.2 =
= 27.8/2 × 7.2 = 13.9 × 7.2 =
= 100.08 m²
What is the area of this triangle in the coordinate plane?
25.0 units²
27.5 units²
50.0 units²
55.0 units²
Answers
Answer:
B.) 27.5²
Explanation:
This should be the triangle per say:
The Area of a Triangle can be found by using the formula- 1/2 x base x height.
In this case: 1/2 x 11 x 5
This equals 27.5².
Hopefully this may help.
A cube of side 3cm contains a sphere within it.if the sphere touches each face of the cube,find the volume of the space between the cube and the sphere
Answers
Answer:
6.86cm^3
Step-by-step explanation:
In order to do this problem you must first find the volume of the cube and subtract the volume of the sphere within it.
Volume of a cube is S^3
using the given 3×3×3= 27cm^3 volume of cube
volume of a sphere is 4/3 pir^3
4/3 × pi x 1.5^3 = 14.14cm^3 volume of sphere
now simply subtract
27-14.14= 6.86 cm^3
Which of these strategies would eliminate a variable in the system of equations? 5x + 3y = 9
4x - 3y = 9 Choose all answers that apply: (Choice A) A Subtract the top equation from the bottom equation. (Choice B) B Add the equations. (Choice C) C Subtract the bottom equation from the top equation.
Answers
Answer:
(Choice B) Add the equations
Step-by-step explanation:
The equations that were given are:
[tex]5x + 3y = 9 \\ 4x - 3y = 9[/tex]
If we add the equequatiothen the y variable would be eliminated:
[tex] +3y + (-3y) = 0[/tex]
A train left a station at noon going at 40 mph. A second train left at 3 pm going in thte same direction at 6o mph. At what time iwll the second train catch up with the first
Answers
Answer:
9PM
Step-by-step explanation:
3pm - 40x3=120 Miles Travelled (Noon Train)
3pm - 60x0=0 Miles Travelled (3pm Train)
4pm - 40x4=160 Miles Travelled (Noon Train)
4pm - 60x1=60 Miles Travelled (3pm Train)
5pm - 40x5=200 Miles Travelled (Noon Train)
5pm - 60x2=120 Miles Travelled (3pm Train)
6pm - 40x6=240 Miles Travelled (Noon Train)
6pm - 60x3=180 Miles Travelled (3pm Train)
7pm - 40x7= 280 Miles Travelled (Noon Train)
7pm - 60x4=240 Miles Travelled (3pm Train)
8pm - 40x8 = 320 Miles Travelled (Noon Train)
8pm - 60x5=300 Miles Travelled (3pm Train)
9pm - 40x9 = 360 Miles Travelled (Noon Train)
9pm - 60x6= 360 Miles Travelled (3pm Train)
giving the following sequence: 21, 24.5, 28 write an equation for the nth term
Answers
Answer:
tₙ = (3.5x + 17.5)
