How do you Use the trapezoidal rule with #n=8# to approximate the integral #int_0^pix^2*sin(x)dx#?
The "trap rule" approximates the area by creating n trapezoids with their bases on the xaxis, top corners along the curve y = f(x), and then adding their areas together.
because the middle terms appear twice, the right side of one trapezoid being the left side of the next. For our example,
You get to evaluate each term to as much accuracy as you need to get your answer to the specified tolerance. Happy calculator plugging!
\another fine answer from dansmath/
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To use the trapezoidal rule with ( n = 8 ) to approximate the integral ( \int_0^{\pi} x^2 \sin(x) , dx ), follow these steps:

Determine the interval width, ( h ), which is given by ( h = \frac{{b  a}}{n} ), where ( n ) is the number of subintervals (in this case, ( n = 8 )) and ( a ) and ( b ) are the lower and upper limits of integration (in this case, ( a = 0 ) and ( b = \pi )).

Calculate ( h ): ( h = \frac{{\pi  0}}{8} = \frac{\pi}{8} ).

Set up the formula for the trapezoidal rule: [ \text{Trapezoidal Rule} = h \left[ \frac{1}{2} f(x_0) + f(x_1) + f(x_2) + \ldots + f(x_{n1}) + \frac{1}{2} f(x_n) \right] ]

Calculate the function values ( f(x_i) ) for each subinterval ( [x_i, x_{i+1}] ). In this case, ( f(x) = x^2 \sin(x) ).

Evaluate the sum using the trapezoidal rule formula.

Multiply the sum by ( h ) to obtain the approximation for the integral.
Following these steps will yield an approximation of the integral ( \int_0^{\pi} x^2 \sin(x) , dx ) using the trapezoidal rule with ( n = 8 ).
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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