How do you use the trapezoidal rule with n = 4 to estimate the integral #int_0^(pi/2)cos(x^2)dx#?

Answer 1

#int_0^(pi/2)cos(x^2)dx~~0.83#

The trapezoidal rule tells us that: #int_b^af(x)dx~~h/2[f(x_0)+f(x_n)+2[f(x_1)+f(x_2)+cdotsf(x_(n-1))]]# where #h=(b-a)/n#
#h=(pi/2-0)/4=pi/8#
So we have: #int_0^(pi/2)cos(x^2)dx~~pi/16[f(0)+f(pi/2)+2[f(pi/8)+f(pi/4)+f((3pi)/8)]]#
#=pi/16[cos((0)^2)+cos((pi/2)^2)+2[cos((pi/8)^2)+cos((pi/4)^2)+cos(((3pi)/8)^2)]]#
#~~pi/16[1-0.78+1.97+1.63+0.36]#
#~~pi/16[4.23]#
#~~0.83#
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Answer 2

To use the trapezoidal rule with ( n = 4 ) to estimate the integral ( \int_0^{\frac{\pi}{2}} \cos(x^2) , dx ), first, divide the interval ( [0, \frac{\pi}{2}] ) into ( n ) equal subintervals. With ( n = 4 ), each subinterval will have a width of ( \frac{\pi}{8} ). Then, compute the function values at the endpoints of these subintervals. Using these function values, calculate the areas of the trapezoids formed by adjacent function values and the width of the subintervals. Finally, sum up these areas to get the approximation of the integral.

Here's the breakdown:

  1. Divide the interval ( [0, \frac{\pi}{2}] ) into ( n = 4 ) subintervals: ( [0, \frac{\pi}{8}], [\frac{\pi}{8}, \frac{\pi}{4}], [\frac{\pi}{4}, \frac{3\pi}{8}], [\frac{3\pi}{8}, \frac{\pi}{2}] ).

  2. Compute the function values at the endpoints of these subintervals: ( f(0) = \cos(0) = 1 ), ( f\left(\frac{\pi}{8}\right) = \cos\left(\frac{\pi^2}{64}\right) ), ( f\left(\frac{\pi}{4}\right) = \cos\left(\frac{\pi^2}{16}\right) ), ( f\left(\frac{3\pi}{8}\right) = \cos\left(\frac{9\pi^2}{64}\right) ), ( f\left(\frac{\pi}{2}\right) = \cos\left(\frac{\pi^2}{4}\right) ).

  3. Calculate the areas of the trapezoids and sum them up: [ \frac{1}{2} \left(1 + \cos\left(\frac{\pi^2}{64}\right)\right) \cdot \frac{\pi}{8} + \frac{1}{2} \left(\cos\left(\frac{\pi^2}{64}\right) + \cos\left(\frac{\pi^2}{16}\right)\right) \cdot \frac{\pi}{8} + \frac{1}{2} \left(\cos\left(\frac{\pi^2}{16}\right) + \cos\left(\frac{9\pi^2}{64}\right)\right) \cdot \frac{\pi}{8} + \frac{1}{2} \left(\cos\left(\frac{9\pi^2}{64}\right) + \cos\left(\frac{\pi^2}{4}\right)\right) \cdot \frac{\pi}{8} ]

    Evaluate this expression to get the approximation of 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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