How do you find the area using the trapezoidal approximation method, given #e^(x^2)#, on the interval [0,1] with n=10?
# int_0^1 e^(x^2) \ dx ~~ 1.4672 # (4dp)
The values of
The Trapezium Rule:
# int_a^bydx ~~ h/2{(y_0+y_n)+2(y_1+y_2+...+y_(n1))}#
uses a series of two consecutive ordinates and a best fit straight line to form trapeziums to approximate the area under a curve, It will have 100% accuracy if
So,
# int_0^1 e^(x^2) \ dx ~~ 0.1/2 { (1 + 2.718281) + #
# " " 2(1.01005 + 1.04081 + 1.094174 + 1.17351 + 1.284025 +#
# " " 1.433329 + 1.632316 + 1.89648 + 2.247907) } #
# " "= 0.05 { + 3.718281 + 2(12.812606) } #
# " "= 0.05 { + 3.718281 + 25.625212 }#
# " "= 0.05 { + 29.343493 }#
# " "= 1.467174#
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To find the area using the trapezoidal approximation method for the function ( e^{x^2} ) on the interval ([0, 1]) with ( n = 10 ), follow these steps:

Divide the interval ([0, 1]) into ( n ) subintervals. Since ( n = 10 ), each subinterval has a width of ( \Delta x = \frac{1  0}{10} = 0.1 ).

Compute the function values at the endpoints of each subinterval and the midpoints of the subintervals. For ( n = 10 ), you will have 11 points: ( x_0 = 0, x_1 = 0.1, x_2 = 0.2, \ldots, x_{10} = 1 ).

Using the trapezoidal rule formula, calculate the area of each trapezoid formed by adjacent points. The formula is: [ \text{Area of trapezoid} = \frac{1}{2} \left[ f(x_i) + f(x_{i+1}) \right] \Delta x ] where ( x_i ) and ( x_{i+1} ) are the endpoints of the subinterval and ( f(x_i) ) and ( f(x_{i+1}) ) are the function values at those points.

Sum up the areas of all the trapezoids to get the total approximate area under the curve.
In this case, with ( n = 10 ), you would calculate the area using the trapezoidal rule with the 11 points obtained from dividing the interval ([0, 1]) into 10 subintervals.
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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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