How do you find the area using the trapezoidal approximation method, given #sqrtx#, on the interval [1,4] with n=3?

Answer 1

# int_1^4 sqrt(x) \ dx = 4.646 # (4dp)

(For comparison the exact value is #4.6666dot6#)

The values of #f(x)=sqrt(x)# are tabulated as follows (using Excel) working to 6dp.

The Trapezium Rule:

# int_a^bydx ~~ h/2{(y_0+y_n)+2(y_1+y_2+...+y_(n-1))}#

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 #y=f(x)# is a straight line and typically provides a good estimate provided #n# is chosen appropriately.

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(8-1) #
# " "= 14/3#
# " "= 4.6666dot6#

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Answer 2

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:

  1. 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 ).

  2. 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.

  3. 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.

  4. 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.

  5. 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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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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