## christinestutes one year ago Use the method of quadrature to estimate the area under the curve and above the x-axis from x = 0 to x = 3. a. 6.3 c. 20.5 b. 10.3 d. 5

1. pooja195

@Michele_Laino

2. christinestutes

3. christinestutes

4. melstutes

I have no idea how to do this?

5. Michele_Laino

6. Michele_Laino

that*

7. Michele_Laino

8. IrishBoy123

"method of quadrature" i looked that up earlier and Wiki describes it as an old/ancient term for what we now call Rieman (sp?) sums. so i guess this is pre-calc numerical solutions for the area under a curve. @melstutes? is that what you are trying to learn?!

9. melstutes

This is a math essentials class. It is a virtual class that is supposed to be an introductory class. I have never had geometry so I am lost with this.

10. Michele_Laino

11. mathmath333

12. melstutes

Thank you mathmath

13. melstutes

Can anyone help solve a different way?

14. melstutes

This is the definition from the class quadrature - The area of an enclosed region on a plane that can be approximated by the sum of the areas of a number of rectangles.

15. mathmath333

there may be way by calculus @Michele_Laino might know, idk much calculus

16. melstutes

I am stumped and don't know where to begin.

17. Nnesha
18. Michele_Laino

I can try to solve using the rectangles approximation method, nevertheless I'm not sure that it is the requested method

19. Michele_Laino

ok! we have to divide the interval into for example three subintervals, as below: $\begin{gathered} {x_0} = 0, \hfill \\ {x_1} = 1 \hfill \\ {x_2} = 2 \hfill \\ {x_3} = 3 \hfill \\ \end{gathered}$

20. Michele_Laino

|dw:1434398306026:dw|

21. Michele_Laino

the requested value is given by the subsequent formula: $I = \frac{{3 - 0}}{3}\left( {f\left( 0 \right) + f\left( 1 \right) + f\left( 2 \right)} \right)$

22. melstutes

Wow, mind blown! I would have NEVER figured that out!

23. Michele_Laino

now we have: $\begin{gathered} f\left( 0 \right) = 5 \hfill \\ f\left( 1 \right) = - \frac{1}{2} + 5 = \frac{9}{2} \hfill \\ f\left( 2 \right) = - \frac{4}{2} + 5 = 3 \hfill \\ \end{gathered}$

24. Michele_Laino

so, after a substitution, we get: $\begin{gathered} I = \frac{{3 - 0}}{3}\left( {f\left( 0 \right) + f\left( 1 \right) + f\left( 2 \right)} \right) = \hfill \\ \hfill \\ = 5 + \frac{9}{2} + 3 = \frac{{25}}{2} = 12.5 \hfill \\ \end{gathered}$

25. Michele_Laino

which is close to 10.3

26. melstutes

Wow. thank you so much for ALL of your time and explanations. I will need to review what you have done and try more examples.

27. Michele_Laino

:)

28. IrishBoy123

https://en.wikipedia.org/wiki/Numerical_integration so " quadrature" = numerical integration

29. anonymous

More importantly, a quadrature is (typically) a trapezoidal approximation that depends on an interpolation of a given function using a specific set of points and a polynomial basis. A good example is the Gaussian quadrature: https://en.wikipedia.org/?title=Gaussian_quadrature which is used to approximate integrals of the form $$\displaystyle\int_{-1}^1f(x)\,dx$$ (and can be extended to approximate integrals over $$[a,b]$$).

30. anonymous

Feel free to skim through chapter 4, there's plenty of information on quadratures. http://ins.sjtu.edu.cn/people/mtang/textbook.pdf

31. phi

I would count the number of "squares" under the curve. If there is part of a square try to estimate to the nearest 1/2 square. The idea is to get a reasonable estimate of the area (not an exact number)

32. anonymous

its d

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