How do you Use the trapezoidal rule with #n=6# to approximate the integral #int_0^3dx/(1+x^2+x^4)dx#?

Answer 1
The answer is #4643/5187#.

The trapezoidal rule is just a formula. From what we are given, we have:

#a=0# #b=3# #n=6# #f(x)=1/(1+x^2+x^4)# #h=(b-a)/n=1/2#

The formula is:

#T=h/2[f(x_0)+2f(x_1)+2f(x_2)+2f(x_3)+2f(x_4)+2f(x_5)+f(x_6)]# #f(x_0)=f(0)=1/1# #f(x_1)=f(1/2)=1/(1+1/4+1/16)=16/21# #f(x_2)=f(1)=1/(1+1+1)=1/3# #f(x_3)=f(3/2)=1/(1+9/4+81/16)=16/133# #f(x_4)=f(2)=1/(1+4+16)=1/21# #f(x_5)=f(5/2)=1/(1+25/4+625/16)=16/741# #f(x_6)=f(3)=1/(1+9+81)=1/91# #T=1/4[1+32/21+2/3+32/133+2/21+32/741+1/91]# #T=4643/5187~~.89512#
Using numeric integration on a graphing calculator, we get #T~~.89537#. So our answer is good for 3 sig figs.
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Answer 2

To use the Trapezoidal Rule with ( n = 6 ) to approximate the integral ( \int_{0}^{3} \frac{dx}{1 + x^2 + x^4} ), you partition the interval ([0, 3]) into ( n = 6 ) subintervals of equal width. Then, you compute the function values at the endpoints of these subintervals and apply the Trapezoidal Rule formula:

[ \int_{a}^{b} f(x) , dx \approx \frac{h}{2} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + \ldots + 2f(x_{n-1}) + f(x_n) \right] ]

where ( h ) is the width of each subinterval, ( a ) and ( b ) are the lower and upper limits of integration respectively, and ( x_0, x_1, \ldots, x_n ) are the partition points.

In this case, ( a = 0 ), ( b = 3 ), ( n = 6 ), and ( h = \frac{b - a}{n} = \frac{3 - 0}{6} = \frac{1}{2} ).

Then, compute the function values at the endpoints of the subintervals:

[ f(0), f(0.5), f(1), f(1.5), f(2), f(2.5), f(3) ]

Substitute these values into the Trapezoidal Rule formula and perform the computation to obtain the approximation for the integral.

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Answer from HIX Tutor

When evaluating a one-sided 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 one-sided 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 one-sided 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 one-sided 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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