How do you use the trapezoidal rule to approximate integral of #e^-3x^2 dx# between [0,1]?

Answer 1

Divide the domain #[0,1]# into smaller trapezoids; calculate the area of each trapezoid; and sum to find an approximation.
Approximate area: # 0.5e^(-3) ~~ 0.24894#

Since #e^(-3)# is a constant and since #int c*f(x) dx = c int f(x) dx# for any constant #c#
I will find an approximate value for #int_0^1 x^2 dx# using trapezoids and then multiply the result by #e^(-3)#
For demonstration purposes I will divide the domain into 4 trapezoidal areas, each with a width of #0.25#
Trapezoid 1 from #0# to #0.25# has an area #=(0.25^2-0^2)/2*0.25 = 0.03125# #color(white)("XXX")#(i.e. average height times width)
Similarly Trapezoid 2 (from #0.25# to #0.5#) #color(white)("XXX")#has an area of #0.09375# Trapezoid 3 (from #0.5# to #0.75#) #color(white)("XXX")#has an area of #0.15625# Trapezoid 4 (from #0.75# to #1.0#) #color(white)("XXX")#has an area of #0.21875#
Total area of all 4 trapezoids #color(white)("XXX")= 0.03125+0.09375+0.15625+0.21875# #color(white)("XXX")=0.5#
So #int_0^1 x^2 dx ~~ 0.5#
and since #int_0^1e^(-3)x^2 dx = e^(-3)int_0^1x^2dx#
#int_0^1 e^(-3)x^2 dx ~~ 0.5e^(-3)#
If you wish to carry this further, note that #e^(-3) ~~0.049787# (according to my calculator)
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Answer 2

To use the trapezoidal rule to approximate the integral of ( e^{-3x^2} ) from ( x = 0 ) to ( x = 1 ), follow these steps:

  1. Divide the interval ([0, 1]) into ( n ) subintervals of equal width. Let's denote the width of each subinterval as ( \Delta x ).
  2. Compute the function values ( y_0, y_1, y_2, ..., y_n ) at the endpoints of each subinterval, where ( y_i = e^{-3x_i^2} ), and ( x_i ) represents the ( i )-th subinterval endpoint.
  3. Apply the trapezoidal rule formula to each subinterval: [ \text{Trapezoidal Rule} = \frac{\Delta x}{2} \left( y_0 + 2y_1 + 2y_2 + \cdots + 2y_{n-1} + y_n \right) ]
  4. Sum up the results from all subintervals to obtain the approximate value of the integral.

If you need further clarification on any step, feel free to ask.

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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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