How do you use the trapezoidal rule with n=9 to approximate the area between the curve #y=x^2 2x +2# from 0 to 3?
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To use the trapezoidal rule with ( n = 9 ) to approximate the area between the curve ( y = x^2  2x + 2 ) from ( x = 0 ) to ( x = 3 ), follow these steps:
 Divide the interval ( [0, 3] ) into ( n = 9 ) subintervals of equal width. Since ( n = 9 ), each subinterval width is ( \Delta x = \frac{3  0}{9} = \frac{1}{3} ).
 Compute the function values at the endpoints of each subinterval. That is, calculate ( y_0 = f(0), y_1 = f(\frac{1}{3}), y_2 = f(\frac{2}{3}), \ldots, y_9 = f(3) ), where ( f(x) = x^2  2x + 2 ).
 Use the trapezoidal rule formula to find the approximate area: [ A \approx \frac{\Delta x}{2} \left[ y_0 + 2(y_1 + y_2 + \ldots + y_8) + y_9 \right] ]
 Plug in the values obtained in steps 1 and 2 into the formula and compute the result.
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To use the trapezoidal rule with ( n = 9 ) to approximate the area between the curve ( y = x^2  2x + 2 ) from ( x = 0 ) to ( x = 3 ), follow these steps:

Determine the interval width, ( \Delta x ), which is given by: [ \Delta x = \frac{b  a}{n} ] where ( a = 0 ) (lower limit), ( b = 3 ) (upper limit), and ( n = 9 ) (number of subintervals).

Calculate ( \Delta x ): [ \Delta x = \frac{3  0}{9} = \frac{1}{3} ]

Create the xvalues for the endpoints of each subinterval. Since ( n = 9 ), there will be 10 points including the endpoints: [ x_0 = 0, , x_1 = \frac{1}{3}, , x_2 = \frac{2}{3}, , x_3 = 1, , x_4 = \frac{4}{3}, , x_5 = \frac{5}{3}, , x_6 = 2, , x_7 = \frac{7}{3}, , x_8 = \frac{8}{3}, , x_9 = 3 ]

Evaluate the function ( y = x^2  2x + 2 ) at each of these xvalues to get the corresponding yvalues.

Use the trapezoidal rule formula to calculate the area of each trapezoid and sum them up: [ A \approx \frac{\Delta x}{2} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + \ldots + 2f(x_{n1}) + f(x_n) \right] ]

Substitute the values into the formula and compute the sum.

The result will be the approximation of the area between the curve ( y = x^2  2x + 2 ) and the xaxis from ( x = 0 ) to ( x = 3 ) using the trapezoidal rule with ( n = 9 ).
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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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