Trapezoidal Rule: Find T8 & M8 for ∫cos(x^2)dx from 1 to 0

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In summary, the Trapezoidal Rule is a method for approximating the value of a definite integral by dividing the area under a curve into trapezoids and calculating the sum of their areas. To find T8 or M8 for a specific integral, the interval must be divided into 8 subintervals and the corresponding formulas must be used. T8 uses trapezoids with equal widths, while M8 uses a combination of equal and unequal widths for more accuracy. The Trapezoidal Rule can be used for any continuous function on the interval, but may not always provide an accurate approximation. Other methods may need to be used for more complex functions.
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pureouchies4717
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hey guys... i kinda forgot how to do this

question is: find the approximations T8 and M8 for integral of cos(x^2)dx from 1 to 0

now my question is whether or not that 8 means that n=8

and also, is this right?

id do the following:

n/2[f(0)+f(.2)+f(.4)+f(.6)+f(.8)+f(1)]

and id use the degree mode on my calculator

is that how i set it up?
 
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arg can someone help me? please
 

1. What is the Trapezoidal Rule?

The Trapezoidal Rule is a method for approximating the value of a definite integral. It divides the area under a curve into trapezoids and calculates the sum of their areas to estimate the integral.

2. How do I find T8 for ∫cos(x^2)dx from 1 to 0?

To find T8, you will need to divide the interval [1,0] into 8 subintervals. Then, use the formula Tn = h/2 [f(x0) + 2f(x1) + 2f(x2) + ... + 2f(xn-1) + f(xn)] to calculate the sum of the areas of the trapezoids. Here, h represents the width of each subinterval and f(x) is the function being integrated.

3. How do I find M8 for ∫cos(x^2)dx from 1 to 0?

To find M8, you will need to divide the interval [1,0] into 8 subintervals as well. Then, use the formula Mn = (h/3) [f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + ... + 2f(xn-2) + 4f(xn-1) + f(xn)] to calculate the sum of the areas of the trapezoids. Again, h represents the width of each subinterval and f(x) is the function being integrated.

4. What is the difference between T8 and M8?

T8 and M8 are both methods for approximating the value of a definite integral using the Trapezoidal Rule. However, T8 uses trapezoids with equal widths, while M8 uses a combination of trapezoids with equal and unequal widths. This results in M8 being a more accurate approximation compared to T8.

5. Can the Trapezoidal Rule be used for any function?

Yes, the Trapezoidal Rule can be used for any function as long as it is continuous on the interval being integrated. However, it may not always provide an accurate approximation and other methods may need to be used for more complex functions.

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