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Simpson's Rule Question

C is a curve with equation y = 3/sqrt(4-4x^2) valid for -1<x<1

R is the region enclosed by the curve and the lines y = 3/2 and y =2

Use Simpson's rule with 4 strips to find an approximation for the area of R.
Give your answer correct to 3 significant figures.

Hiya, I'm just wondering how anyone would approach the question above if you were solving it - It seems a little strange. If anyone can attach written responses that's amazing but absolutely don't feel inclined to! :smile:

Reply 1

Nitrotoluene

Original post

by イロハ

C is a curve with equation y = 3/sqrt(4-4x^2) valid for -1<x<1
R is the region enclosed by the curve and the lines y = 3/2 and y =2
Use Simpson's rule with 4 strips to find an approximation for the area of R.
Give your answer correct to 3 significant figures.
Hiya, I'm just wondering how anyone would approach the question above if you were solving it - It seems a little strange. If anyone can attach written responses that's amazing but absolutely don't feel inclined to! :smile:

Let's discuss how to solve this problem; I am not doing the calculations directly.
First, make sure you have a good understanding of region R.
There's curve C, which is enclosed by the horizontal lines y=3/2 and y=2.
Here's a suggested approach:
Region Visualization:
What is the shape of the curve y=3/ (4−4x^2)?
Think about its domain and symmetries.
Draw the lines y=2 and y=3/2
Determine where these lines intersect curve C.
These points of intersection are the limits of integration for your area calculation.
Solve for x when y=3/2 and when y=2.
Area Calculation Reassessment:
Simpson's rule is for the approximation of the area beneath a curve, between the curve and the x-axis.
Region R is bounded by the curve and horizontal lines.
It will perhaps be easier to integrate with respect to y than x. If integrating with respect to y, write x in terms of y. Rewrite the equation y=3/(4−4x^2) to isolate x.
Keep in mind that x can have a positive and negative root.
Think about the width of the region at various y values. If you're integrating with respect to y, the height of strips will be Δy.
Each strip's length will be the difference between the rightmost and leftmost x values for a given y.
Simpson's Rule Setup:
Once you've chosen to integrate with respect to x or y, determine the limits of integration, the coordinates at which the boundaries of the region intersect.You will need to use 4 strips. Consider how the number of strips relates to the number of ordinates (or data points) for Simpson's Rule.
Calculate each strip's width (Δx or Δy).
Identify the particular x (or y) values that are required to evaluate your function.
Recall Simpson's Rule formula.
Applying Simpson's Rule (Conceptually): At every necessary point, calculate the corresponding value of your function (either x or y).
Plug in these values in Simpson's Rule formula.
Key questions:
What are the x-coordinates where the curve meets y=3/ 2
What are the x-coordinates of where the curve intersects y=2?
Is it easier to integrate y with respect to x, or x with respect to y?
Consider the shape of the region.If opting to integrate with respect to y, what is the new function going to be (i.e., x=f(y))?
What are the lower and upper integration bounds when integrating with respect to y?
How many ordinates are required by 4 strips for Simpson's Rule?
What is the width of each strip, and what are the particular y-values (or x-values, if you prefer) that you will use for the calculation of Simpson's Rule?
Do take the time to go through these steps.
The goal is that you perform each part of the process, including any algebraic rearrangements and evaluations that are required, to arrive at the answer. IMHO...of course!😀
I know you can!💫

Ciao from Italy,
Sandro

(edited 9 months ago)

Reply 2

イロハ

Thank you Sandro! My thought process was to reflect in the line y = x but doing this means that y = 3/2 and y = 2 now becomes x = 3/2 and x = 2 and apply simpson's rule - but double this to give the full area due to symmetry - not sure if this is how anyone else would do it but I'm getting a small answer so I'm not sure!

Happy Studying.

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