How do you Use the trapezoidal rule with #n=6# to approximate the integral #int_0^1e^-sqrt(x)dx#?

Answer 1
Let #f(x)=e^{-sqrt{x}}#.

The definite integral

#int_0^1 f(x) dx#

can be approximated by Trapezoid Rule

#approx T_6=[f(0/6)+2f(1/6)+2f(2/6)+2f(3/6)+2f(4/6)+2f(5/6)+f(6/6)]{1-0}/{2(6)}approx 0.54#
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Answer 2

Using the trapezoidal rule with ( n = 6 ) to approximate the integral ( \int_0^1 e^{-\sqrt{x}} , dx ) involves dividing the interval [0, 1] into 6 subintervals of equal width, computing the function values at the endpoints of these subintervals, and then applying the trapezoidal rule formula.

The formula for the trapezoidal rule with ( n ) subintervals is:

[ \int_a^b f(x) , dx \approx \frac{h}{2} \left[ f(x_0) + 2\sum_{i=1}^{n-1} f(x_i) + f(x_n) \right] ]

where:

  • ( h = \frac{b - a}{n} ) is the width of each subinterval,
  • ( x_i = a + ih ) for ( i = 0, 1, 2, \ldots, n ) are the endpoints of the subintervals,
  • ( f(x_i) ) are the function values at the endpoints.

For ( n = 6 ), we have ( h = \frac{1 - 0}{6} = \frac{1}{6} ). The endpoints are ( x_0 = 0, x_1 = \frac{1}{6}, x_2 = \frac{2}{6}, x_3 = \frac{3}{6}, x_4 = \frac{4}{6}, x_5 = \frac{5}{6}, x_6 = 1 ).

Calculating the function values at these points:

  • ( f(x_0) = e^{-\sqrt{0}} = 1 ),
  • ( f(x_1) = e^{-\sqrt{\frac{1}{6}}} ),
  • ( f(x_2) = e^{-\sqrt{\frac{2}{6}}} ),
  • ( f(x_3) = e^{-\sqrt{\frac{3}{6}}} ),
  • ( f(x_4) = e^{-\sqrt{\frac{4}{6}}} ),
  • ( f(x_5) = e^{-\sqrt{\frac{5}{6}}} ),
  • ( f(x_6) = e^{-\sqrt{1}} = e^{-1} ).

Now, we can plug these values into the trapezoidal rule formula and compute the approximation.

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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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