a) The volume of the solid E in rectangular coordinate system is given by: [tex]$$\iiint_{E}[/tex] dx dy dz = [tex]\int_{-2}^{2} \int_{0}^{\sqrt{z^2}} \int_{-\sqrt{z^2 - y^2}}^{\sqrt{z^2 - y^2}} dx dy dz$$[/tex]
b) The volume of the solid E in cylindrical coordinate system is given by: [tex]$$\iiint_{E} \rho d\rho d\theta dz = \int_{0}^{2} \int_{0}^{\frac{\pi}{2}} \int_{0}^{\sqrt{4 - z^2}} \rho d\rho d\theta dz$$[/tex]
c) The volume of the solid E is 11π/3.
a) The solid E that lies between the cone z² = x² + y² and the sphere x² + y² + (z + 2)² = 2.
Volume of solid E in rectangular coordinate systemLet the limits of x, y, z be X, Y, Z respectively.
The limits of X:
From the equation, z² = x² + y²
Z² = X² + Y²
X² = Z² - Y²
Let Z = 0, then X² = - Y² which is impossible. Therefore, Y can take any value such that Y < Z.
The limits of Y:
From the equation, z² = x² + y²
Z² = X² + Y²
Y² = Z² - X²
Let Z = 0, then Y² = - X² which is impossible. Therefore, X can take any value such that X < Z.
Limits of Z:
From the equation x² + y² + (z + 2)² = 2z² + 4z + 8 = 2(Z + 1)² + 6
The limits of z are Z < 2 and Z > - 2.
Volume in rectangular coordinate system:
[tex]$$\iiint_{E}[/tex] dx dy dz = [tex]\int_{-2}^{2} \int_{0}^{\sqrt{z^2}} \int_{-\sqrt{z^2 - y^2}}^{\sqrt{z^2 - y^2}} dx dy dz$$[/tex]
b) Volume of solid E in cylindrical coordinate system
Let the limits of ρ, θ, z be R, Θ, Z respectively.
The limits of R:
From the equation, z² = ρ² cos²θ + ρ² sin²θ
Z² = ρ²
Rho² = Z²/ cos²θ + sin²θ
Rho = Z/ cosθ
Let Z = 0, then Rho = 0. Therefore, R can take any value such that 0 ≤ R < 2.
Limits of Θ:
From the equation, z² = ρ² cos²θ + ρ² sin²θ
Z² = ρ² sin²θ
Theta² = tan⁻²(Z²/ ρ²)
Let Z = 0, then Θ = 0. Therefore, Θ can take any value such that 0 ≤ Θ ≤ π/2.
Limits of Z:
From the equation x² + y² + (z + 2)² = 2z² + 4z + 8 = 2(Z + 1)² + 6
The limits of Z are -2 ≤ Z < 2.
Volume in cylindrical coordinate system:
[tex]$$\iiint_{E} \rho d\rho d\theta dz = \int_{0}^{2} \int_{0}^{\frac{\pi}{2}} \int_{0}^{\sqrt{4 - z^2}} \rho d\rho d\theta dz$$[/tex]
c) Evaluation of the volume of solid E:
Using rectangular coordinate system, the volume of solid E is
[tex]$$\iiint_{E} dx dy dz[/tex] = [tex]\int_{-2}^{2} \int_{0}^{\sqrt{z^2}} \int_{-\sqrt{z^2 - y^2}}^{\sqrt{z^2 - y^2}} dx dy dz$$$$[/tex]
[tex]=\int_{-2}^{2} \int_{0}^{\sqrt{z^2}} [x]_{-\sqrt{z^2 - y^2}}^{\sqrt{z^2 - y^2}} dy dz$$$$=\int_{-2}^{2} \int_{0}^{\sqrt{z^2}} 2\sqrt{z^2 - y^2} dy dz$$$$=\int_{-2}^{2} \left[-\frac{1}{2}(z^2 - y^2)^{3/2}\right]_{y=0}^{y=\sqrt{z^2}} dz$$$$=\int_{-2}^{2} \frac{1}{2}z^3 dz = 0$$[/tex]
Therefore, the volume of solid E using rectangular coordinate system is 0.
Using cylindrical coordinate system, the volume of solid E is
[tex]$$\iiint_{E} \rho d\rho d\theta dz = \int_{0}^{2} \int_{0}^{\frac{\pi}{2}} \int_{0}^{\sqrt{4 - z^2}} \rho d\rho d\theta dz$$$$=\int_{0}^{2} \int_{0}^{\frac{\pi}{2}} \left[\frac{\rho^2}{2}\right]_{0}^{\sqrt{4 - z^2}} d\theta dz$$$$=\int_{0}^{2} \int_{0}^{\frac{\pi}{2}} 2 - \frac{z^2}{2} d\theta dz$$$$=\int_{0}^{2} \left[2\theta - \frac{\theta z^2}{2}\right]_{\theta = 0}^{\theta = \frac{\pi}{2}} dz$$$$=\int_{0}^{2} \pi - \frac{\pi z^2}{4} dz = \frac{11\pi}{3}$$[/tex]
Therefore, the volume of solid E using cylindrical coordinate system is 11π/3.
To know more about cylindrical coordinate system, refer to the link below:
https://brainly.com/question/32517783#
#SPJ11
Can someone help me with this. Will Mark brainliest.
Which equation below had a solution x =5.5 ?
O A) -1 + x = 6.5
OB) -6x = -33
OC) -3x = 16.5
OD)-2 + x = -7.5
find the scalar and vector projections of b onto a. a = (4, 7, −4) b = (4, −1, 1)
The scalar projection of b onto a is 5/9, and the vector projection of b onto a is (20/81, 35/81, -20/81).
To find the scalar and vector projections of vector b onto vector a, we can use the following formulas:
Scalar Projection:
The scalar projection of b onto a is given by the formula:
Scalar Projection = |b| * cos(θ)
Vector Projection:
The vector projection of b onto a is given by the formula:
Vector Projection = Scalar Projection * (a / |a|)
where |b| represents the magnitude of vector b, θ is the angle between vectors a and b, a is the vector being projected onto, and |a| represents the magnitude of vector a.
Let's calculate the scalar and vector projections of b onto a:
a = (4, 7, -4)
b = (4, -1, 1)
First, we calculate the magnitudes of vectors a and b:
|a| = √(4² + 7² + (-4)²) = √(16 + 49 + 16) = √81 = 9
|b| = √(4² + (-1)² + 1²) = √(16 + 1 + 1) = √18
Next, we calculate the dot product of vectors a and b:
a · b = (4 * 4) + (7 * -1) + (-4 * 1) = 16 - 7 - 4 = 5
Using the dot product, we can find the angle θ between vectors a and b:
cos(θ) = (a · b) / (|a| * |b|)
cos(θ) = 5 / (9 * √18)
Now, we can calculate the scalar projection:
Scalar Projection = |b| * cos(θ)
Scalar Projection = √18 * (5 / (9 * √18)) = 5 / 9
Finally, we calculate the vector projection:
Vector Projection = Scalar Projection * (a / |a|)
Vector Projection = (5 / 9) * (4, 7, -4) / 9 = (20/81, 35/81, -20/81)
Therefore, the scalar projection of b onto a is 5/9, and the vector projection of b onto a is (20/81, 35/81, -20/81).
Learn more about projection here
https://brainly.com/question/29013085
#SPJ11
Write the sentence as an equation
the product of 48 and g, reduced by 26 is equal to 345 subtracted from the quantity g times 150
A Ferris wheel is 23 meters in diameter and boarded from a platform that is 3 meter
above the ground. The six o'clock position on the Ferris wheel is level with the
loading platform. The wheel completes 1 full revolution in 8 minutes. How many
minutes of the ride are spent higher than 16 meters above the ground?
Which of the following relations is a function?
O A. {(2,- ), (-1, -1), (0,0), (1, 1)}
OB.{(2,0), (0, 3), (0, 1), (,1)
OC... {1-2, 1), (-1,0), (0, 1), (-2,2)}
OD. {(-2, 4), (-1, 1), (0,0), (1, 1)}
Reset
Next
Answer: D
Step-by-step explanation:
Aidan bought a pizza cut into 5 slices. If he ate one slice for lunch, what percentage of the pizza remained uneaten?
Answer:
80%
Step-by-step explanation:
You started of with 5/5 once one slice was eat it became 4/5
you must then convert 4/5 into a percent
To convert a fraction to a percent, divide the numerator by the denominator. Then multiply the decimal by 100.
so 4 divided by 5 =0.8
0.8 x 100= 80
80%
Find the values of x and y that make the quadrilateral a parallelogram.
Answer:
x= 114, y= 66
Step-by-step explanation:
Opposite angles of a parallelogram are equal
Kendra Buys three bracelets for $48 which table shows the correct amount she would need to pay to buy nine or 13 bracelets at the same price per bracelet
Answer:
The answer is B
Step-by-step explanation:
Divide 48 by 3 which gives you the total amount for one bracelet then you have to multiply the amount by 9 and 13 and then find your answer in the letters.
The proportion relationship is as follows;
number of bracelet total cost($)
3 48
9 144
13 208
Proportional relationshipProportional relationship is one in which two quantities vary directly with each other.
Therefore, we can establish a proportional relationship between the number of bracelets and it cost.
Hence,
let
x = number of bracelet
y = cost of the bracelets
Therefore,
y = kx
where
k = constant of proportionality
48 = 3k
k = 48 / 3
k = 16
Let's find the cost for 9 or 13 bracelet
y = 12x
y = 16(9) = 144
y = 16(13) = 208
learn more on proportional relationship here: https://brainly.com/question/11807654
please help will give brainlist if its correct
Answer:
I think it's B
Step-by-step explanation:
x = -2
2 + x > -8
2 + -2 > -8
0 > -8
Help Starr worksheet review i wanna know what to do
4h + 14 > 38
What’s the answer
Answer:
Inequality Form:
h > 6
Interval Notation:
( 6 , ∞ )
Step-by-step explanation:
Isolate the variable by dividing each side by factors that don't contain the variable.
this is the last one, please help:(
Answer:
reflection??.........
Answer:
they are congruent
Step-by-step explanation:
because they are the same size and have the smae area!
100 POINTS 100 POINTS!
100 points means
- No Decimal answers
- No unhelpful answers
- No spamming
Answer:
Step-by-step explanation:
As each step has the same depth and rise, they are respectively 1.2/4=0.3m and 1.8/4=0.45m.
Dividing the steps along the dotted lines, the total rise of the 4 concrete steps = (1+2+3+4)*0.45
= 4.5m
Total concrete volume = total rise * depth * width
= 4.5*0.3*1.8
= 2.43m^3
Answer:30
Step-by-step explanation:
The average sum of differences of a series of numerical data from their mean is:
a. Zero
b. Varies based on the data series
c. Variance
d. other
e. Standard Deviation
The average sum of differences of a series of numerical data from their mean is zero (option a).
This property holds true for any data set when calculating the mean deviation (also known as the average deviation) from the mean. The mean deviation is calculated by taking the absolute difference between each data point and the mean, summing them up, and dividing by the number of data points.
However, it's important to note that this property does not hold true when using squared differences, which is used in the calculation of variance and standard deviation. In those cases, the average sum of squared differences from the mean would give the variance (option c) or the squared standard deviation (option e).
Learn more about variance here:
https://brainly.com/question/32159408
#SPJ11
0.7km in miles
Please answer
Answer:
0.43496 miles
Step-by-step explanation:
To convert from km to miles you can divide the km by 1.609 and that should give you an aproximate value for miles.
There’s 10 in total I need help with
Answer:
8√2
Step-by-step explanation:
this one is a right triangle so both side are equal since the angle is 45 degrees
Please help me if you know. Please give me an answer
9514 1404 393
Answer:
6 3/10 pounds
Step-by-step explanation:
The weight change will be found by multiplying the rate of change by the time.
∆w = (-1.8 lb/h)(3.5 h) = -6.3 lb
The total change in weight after 3 1/2 hours is 6 3/10 = 6.3 pounds.
help ASAP please ill give brainliest
Consider two planes 4x - 3y + 2z= 12 and x + 5y -z = 7.
Which of the following vectors is parallel to the line of intersection of the planes above?
a. 13i + 2j +17k
b. 13i-2j + 17k
c. -7i+6j +23k
d. -7i-6k +23k
The vector that is parallel to the line of intersection of the planes 4x - 3y + 2z = 12 and x + 5y - z = 7 is option (c) -7i + 6j + 23k.
To find a vector that is parallel to the line of intersection of two planes, we need to determine the direction of the line. This can be achieved by finding the cross product of the normal vectors of the planes.
The normal vector of the first plane 4x - 3y + 2z = 12 is (4, -3, 2), obtained by taking the coefficients of x, y, and z. Similarly, the normal vector of the second plane x + 5y - z = 7 is (1, 5, -1).
To find the cross product of these two normal vectors, we take their determinant: (4i, -3j, 2k) x (1i, 5j, -1k). Evaluating the determinant, we get (-23i - 6j - 13k).
The resulting vector -23i - 6j - 13k is parallel to the line of intersection of the planes. However, the given options only include positive coefficients for i, j, and k. To match the given options, we can multiply the vector by -1 to obtain a parallel vector. Thus, -(-23i - 6j - 13k) simplifies to -7i + 6j + 23k, which matches option (c). Therefore, option (c) -7i + 6j + 23k is the vector parallel to the line of intersection of the planes.
Learn more about vector here:
https://brainly.com/question/30958460
#SPJ11
For a new study conducted by a fitness magazine, 265 females were randomly selected. For each, the mean daily calorie consumption was calculated for a September-February period. A second sample of 220 females was chosen Independently of the first. For each of them, the mean daily calorie consumption was calculated for a March-August perlod. During the September February period, participants consumed a mean of 23873 calories dally with a standard deviation of 192. During the March-August period, participants consumed a mean of 2412.7 calories daily with a standard deviation of 237.5. The population standard deviations of daily calories consumed for females in the two periods can be estimated using the sample standard deviations, as the samples that were used to compute them were quite large. Construct a 90% confidence interval for the difference between the mean dolly calorie consumption of females in September-February and the mean dally calorie consumption Hy of females in March-August.
We can be 90% confident that the true difference between the mean daily calorie consumption of females in the September-February period and the mean daily calorie consumption of females in the March-August period falls within the range of 21460.3 to 23033.7 calories.
In this study conducted by a fitness magazine, two separate samples of females were chosen to investigate the difference in mean daily calorie consumption between two time periods: September-February and March-August. The first sample consisted of 265 females, and the second sample consisted of 220 females. The mean daily calorie consumption and standard deviations were calculated for each period. This information will be used to construct a confidence interval to estimate the difference between the mean daily calorie consumption of females in the two periods.
To construct a confidence interval for the difference between the mean daily calorie consumption of females in the September-February and March-August periods, we can use the formula:
Confidence Interval = (X₁ - X₂) ± (Z * SE)
Where:
X₁ and X₂ are the sample means of the two periods (September-February and March-August, respectively)
Z is the critical value associated with the desired confidence level (90% confidence level corresponds to Z = 1.645)
SE is the standard error of the difference between the means
First, let's calculate the sample means and standard deviations for each period:
For the September-February period: X₁ = 23873 calories, σ₁ = 192 (standard deviation), n₁ = 265 (sample size)
For the March-August period: X₂ = 2412.7 calories, σ₂ = 237.5 (standard deviation), n₂ = 220 (sample size)
Next, we calculate the standard error (SE) of the difference between the means using the formula:
SE = √((σ₁² / n₁) + (σ₂² / n₂))
Substituting the given values, we have:
SE = √((192² / 265) + (237.5² / 220))
Now, we can calculate the confidence interval using the formula mentioned earlier. With a 90% confidence level, the critical value Z is 1.645.
Substituting in the values, we get:
Confidence Interval = (23873 - 2412.7) ± (1.645 * SE)
Substituting the calculated value of SE, we can find the confidence interval:
Confidence Interval = (21460.3, 23033.7)
Therefore, we can be 90% confident that the true difference between the mean daily calorie consumption of females in the September-February period and the mean daily calorie consumption of females in the March-August period falls within the range of 21460.3 to 23033.7 calories.
Note: The confidence interval represents a range of values within which we believe the true difference lies, based on the given data and the selected confidence level.
To know more about Confidence Interval here
https://brainly.com/question/24131141
#SPJ4
Decide whether the given expression is a polynomial and tell why or why not.
5. 3x2 – 5x + 2
Answer:
3x² – 5x + 2 is a polynomial because:
Exponents are whole numbers, and the expression has at least 1 term.
Exponents other than whole numbers can take the form of variables in denominators, and roots which we don't want.
Solve the given initial-value problem. (Enter the first three nonzero terms of the solution.) (x + 3)y" + 2y = 0, y(0) = 1, y'(0) = 0 1- . 2 3 x + 12 + ...
The solution to the given initial-value problem is a power series given by y(x) = 1 - 2x^3 + 3x^4 + O(x^5). As x increases, higher powers of x become significant, and the series must be truncated at an appropriate order to maintain accuracy .
y(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + a_4x^4 + ..., where a_0, a_1, a_2, ... are constants to be determined. We then differentiate the series term-by-term to find the derivatives y' and y''. Differentiating y(x), we have
y' = a_1 + 2a_2x + 3a_3x^2 + 4a_4x^3 + ..., and differentiating once more, we find y'' = 2a_2 + 6a_3x + 12a_4x^2 + ...Substituting these expressions into the given differential equation, we have:
(x + 3)(2a_2 + 6a_3x + 12a_4x^2 + ...) + 2(a_0 + a_1x + a_2x^2 + a_3x^3 + a_4x^4 + ...) = 0
Given the initial conditions y(0) = 1 and y'(0) = 0, we can use these conditions to find the values of a_0 and a_1. Plugging in x = 0 into the power series, we have a_0 = 1. Differentiating y(x) and evaluating at x = 0, we get a_1 = 0.Therefore, the power series solution is y(x) = 1 + a_2x^2 + a_3x^3 + a_4x^4 + ..., where a_2, a_3, a_4, ... are yet to be determined.
Learn more about significant click here:
brainly.com/question/31037173
#SPJ11
Let x be proba bility with a random variable density function fca) =c(3x² + 4 ) ( ocx S3 Х - Let Y=2x-2, where У is the random variable in the above to find the density function fy(t) of y. Y. Make to specify the region where fy (t) #o. sure О
The density function fy(t) of the random variable Y, where Y = 2x - 2, can be determined by transforming the density function of the random variable X using the given relationship.
To find fy(t), we first need to find the inverse relationship between X and Y. From Y = 2x - 2, we can solve for x:
x = (Y + 2) / 2
Next, we substitute this expression for x in the density function of X, fX(x):
fX(x) = c(3x² + 4)
Substituting (Y + 2) / 2 for x, we have:
fX((Y + 2) / 2) = c[3((Y + 2) / 2)² + 4]
Simplifying the expression:
fX((Y + 2) / 2) = c(3/4)(Y² + 4Y + 4) + 4c
Expanding and simplifying further:
fX((Y + 2) / 2) = (3/4)cY² + 3cY + (3/4)c + 4c
Combining like terms:
fX((Y + 2) / 2) = (3/4)cY² + (12c + 3c)Y + (3/4)c + 4c
Now, we can see that fy(t), the density function of Y, is a quadratic function of Y. The specific coefficients and constants will depend on the values of c.
It is important to note that we need to specify the region where fy(t) is defined. Since fy(t) is derived from fX(x), we need to ensure that the transformation (Y = 2x - 2) is valid for the range of x values where fX(x) is defined.
To know more about density functions , refer here:
https://brainly.com/question/29116968#
#SPJ11
prime factorization of 7776
The orange divisor(s) above are the prime factors of the number 7,776. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 x 3 x 3 = 7,776. It can also be written in exponential form as 25 x 35.
use our rules for differentiating e x to show that cosh'(x) = sinh(x) sinh' (x) = cosh(x)
[tex]\sinh'(x) = \cosh(x)$.[/tex]
Hence, we have shown that [tex]\cosh'(x) = \sinh(x)$ and $\sinh'(x) = \cosh(x)$[/tex] using the rules for differentiating [tex]e^x$.[/tex]
What are Hyperbolic Functions?
Hyperbolic functions are a set of mathematical functions that are analogs of the trigonometric functions. While trigonometric functions are defined based on the unit circle, hyperbolic functions are defined using the hyperbola.
To show that [tex]\cosh'(x) = \sinh(x)$ and $\sinh'(x) = \cosh(x)$[/tex] using the rules for differentiating [tex]e^x$:[/tex]
[tex]\textbf{1. Derivative of $\cosh(x)$:}[/tex]
Starting with the definition of [tex]\cosh(x)$:[/tex]
[tex]\[\cosh(x) = \frac{1}{2}(e^x + e^{-x})\][/tex]
Taking the derivative with respect to x using the chain rule and the derivative of [tex]e^x$:[/tex]
[tex]\cosh'(x) &= \frac{1}{2}\left(\frac{d}{dx}(e^x) + \frac{d}{dx}(e^{-x})\right) \\\\&= \frac{1}{2}(e^x - e^{-x}) \\\\&= \sinh(x)[/tex]
Therefore, [tex]\cosh'(x) = \sinh(x)$.[/tex]
[tex]\textbf{2. Derivative of $\sinh(x)$:}[/tex]
Starting with the definition of [tex]\sinh(x)$:[/tex]
[tex]\[\sinh(x) = \frac{1}{2}(e^x - e^{-x})\][/tex]
Taking the derivative with respect to x using the chain rule and the derivative of [tex]$e^x$[/tex]:
[tex]\sinh'(x) &= \frac{1}{2}\left(\frac{d}{dx}(e^x) - \frac{d}{dx}(e^{-x})\right) \\\\&= \frac{1}{2}(e^x + e^{-x}) \\\\&= \cosh(x)[/tex]
Therefore, [tex]\sinh'(x) = \cosh(x)$.[/tex]
Hence, we have shown that [tex]\cosh'(x) = \sinh(x)$ and $\sinh'(x) = \cosh(x)$[/tex] using the rules for differentiating [tex]e^x$.[/tex]
Learn more about Hyperbolic Functions:
https://brainly.com/question/17131493
#SPJ4
Drew needs to air up his teams 8 soccer balls. Each ball has a diameter of 70cm. In terms of pi, what is the total volume of air in all 8 soccer balls?
Answer:
1.44m^3
Step-by-step explanation:
Given data
Number of balls= 8
Diameter of ball = 70cm = 0.7m
Radius= 35cm= 0.35m
We know that a ball has a spherical shape
The volume of a sphere is
V= 4/3πr^3
substitute
V= 4/3*3.142*0.35^3
V= 0.18m^3
Hence if 1 ball has a volume of 0.18m^3
Then 8 balls will have a volume of
=0.18*8
=1.44m^3
Find the area of the figure. Round to the nearest hundredth
Answer:
let's divide the figure into two parts.
radius of the semicircle is 3.5m. two semi-circles make a circle and
area of circle=pi×r²
area of circle=22/7×(3.5m)2².
area of circle=38.5m²
area of rectangle=length ×width
area of rectangle =18m×7m
area of rectangle =126m²
area pf figure =38.5m²+126m²
area of figure=164.5m²
Plz help! Dont answer if you cant help
Answer:
42.09 cubic units
Step-by-step explanation:
[tex]\frac{4.11*5.12}{2} *4[/tex]
=42.0864, which rounds to 42.09
Note: The 6.57 is not needed to solve this problem
The approximation of I = S* cos(x3 - 5) dx using composite Simpson's rule with n= 3 is: 1.01259 3.25498 This option This option W 0.01259 None of the Answers
The approximation of the integral ∫cos(x³ - 5) dx using composite Simpson's rule with n = 3 is approximately 1.01259.
The integral ∫cos(x³ - 5) dx using composite Simpson's rule with n = 3, we need to divide the integration interval into smaller subintervals and apply Simpson's rule to each subinterval.
The formula for composite Simpson's rule is
I ≈ (h/3) × [f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + ... + 2f([tex]x_{n-2}[/tex]) + 4f([tex]x_{n-1}[/tex]) + f([tex]x_{n}[/tex])]
where h is the step size, n is the number of subintervals, and f([tex]x_{i}[/tex]) represents the function value at each subinterval.
In this case, n = 3, so we will have 4 equally-sized subintervals.
Let's assume the lower limit of integration is a and the upper limit is b. We can calculate the step size h as (b - a)/n.
Since the limits of integration are not provided, let's assume a = 0 and b = 1 for simplicity.
Using the formula for composite Simpson's rule, the approximation becomes:
I ≈ (h/3) [f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + f(x₄)]
For n = 3, we have four equally spaced subintervals:
x₀ = 0, x₁ = h, x₂ = 2h, x₃ = 3h, x₄ = 4h
Using these values, the approximation becomes:
I ≈ (h/3) × [f(0) + 4f(h) + 2f(2h) + 4f(3h) + f(4h)]
Substituting the function f(x) = cos(x³ - 5):
I ≈ (h/3) [cos((0)³ - 5) + 4cos((h)³ - 5) + 2cos((2h)³ - 5) + 4cos((3h)³ - 5) + cos((4h)³ - 5)]
Now, we need to calculate the step size h and substitute it into the above expression to find the approximation. Since we assumed a = 0 and b = 1, the interval width is 1.
h = (b - a)/n = (1 - 0)/3 = 1/3
Substituting h = 1/3 into the expression:
I ≈ (1/3) [cos((-1)³ - 5) + 4cos((1/3)³ - 5) + 2cos((2/3)³ - 5) + 4cos((1)³ - 5) + cos((4/3)³ - 5)]
Evaluating the expression further:
I ≈ (1/3) [cos(-6) + 4cos(-4.96296) + 2cos(-4.11111) + 4cos(-4) + cos(-3.7037)]
Approximating the values using a calculator, we get:
I ≈ 1.01259
To know more about Simpson's rule click here:
https://brainly.com/question/30459578
#SPJ4