How do you find the area using the trapezoidal approximation method, given #sqrtx#, on the interval [1,4] with n=3?
# int_1^4 sqrt(x) \ dx = 4.646 # (4dp)(For comparison the exact value is
#4.6666dot6# )
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_1^4 sqrt(x) \ dx = 1/2 { (1 + 2) + 2(1.414213 + 1.73205) } #
# " "= 0.5 { + 3 + 2(3.146264) } #
# " "= 0.5 { + 3 + 6.292528 } #
# " "= 0.5 { + 9.292528 } #
# " "= 4.646264 #
Exact Value:
# int_1^4 sqrt(x) \ dx = [2/3x^(3/2)]_1^4 #
# " "=2/3{4^(3/2)1^(3/2)} #
# " "= 2/3(81) #
# " "= 14/3#
# " "= 4.6666dot6#
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To find the area using the trapezoidal approximation method with ( n = 3 ) on the interval ([1, 4]) for the function ( \sqrt{x} ), follow these steps:

Divide the interval ([1, 4]) into ( n ) subintervals of equal width. Since ( n = 3 ), the width of each subinterval is ( \Delta x = \frac{4  1}{3} = 1 ).

Determine the function values at the endpoints of each subinterval. For ( n = 3 ), this involves evaluating ( \sqrt{x} ) at ( x = 1, 2, 3, ) and ( 4 ). These values are ( \sqrt{1} = 1, \sqrt{2}, \sqrt{3}, ) and ( \sqrt{4} = 2 ), respectively.

Use the trapezoidal rule formula to calculate the area under the curve within each subinterval and then sum these areas. The formula for the area of one trapezoid is ( \frac{1}{2}(b_1 + b_2)h ), where ( b_1 ) and ( b_2 ) are the lengths of the two parallel sides (the function values at the endpoints of the subinterval), and ( h ) is the width of the subinterval.

For ( n = 3 ), there are three subintervals: ([1, 2], [2, 3], ) and ( [3, 4] ). Calculate the area of each trapezoid using the formula from step 3.

Sum the areas of all three trapezoids to find the approximate area under the curve within the interval ([1, 4]).
Following these steps, you can use the trapezoidal approximation method to find the area under the curve of ( \sqrt{x} ) on the interval ([1, 4]) with ( n = 3 ).
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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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