How do you find the area using the trapezoidal approximation method, given #cos(4 x) dx#, on the interval [1, 2] with n=10?
# int_(1)^2 \ cos4x \ dx ~~ 0.0510 # 4dp
We have:
# y = cos(4x) #
We want to estimate
# Deltax = (2(1))/10 = 0.3#
The values of the function are tabulated as follows;
Trapezium Rule
# A = 0.3/2 * { 0.653644  0.1455 + #
# \ \ \ \ \ \ \ \ \ 2*(0.942222  0.0292 + 0.921061 + #
# \ \ \ \ \ \ \ \ \ 0.696707  0.416147  0.998295  #
# \ \ \ \ \ \ \ \ \ 0.307333 + 0.775566 + 0.869397) } #
# \ \ \ = 0.15 * { 0.799144 + 2*(0.569535) } #
# \ \ \ = 0.15 * { 0.799144 + 1.139069 } #
# \ \ \ = 0.15 * 0.339926 #
# \ \ \ = 0.050989 #
Actual Value
For comparison of accuracy:
# A= int_(1)^2 \ cos4x \ dx #
# " \ \ \ = [1/4sin4x]_(1)^2 #
# " \ \ \ = 1/4(sin8sin(4)) \ \ \ \ # (radians!)
# " \ \ \ = 1/4(0.989358...  0.756802 ...) #
# " \ \ \ = 0.058138 ... #
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To find the area using the trapezoidal approximation method with ( n = 10 ) for the function ( \cos(4x) ) on the interval ([1, 2]), follow these steps:

Divide the interval ([1, 2]) into ( n ) subintervals of equal width. Since ( n = 10 ), each subinterval width would be ( \Delta x = \frac{2  (1)}{10} = \frac{3}{10} ).

Compute the function values at the endpoints of each subinterval, including the endpoints of the interval ([1, 2]). This means you'll compute ( \cos(4x) ) at ( x = 1, 1 + \Delta x, 1 + 2\Delta x, \ldots, 2 ).

Apply the trapezoidal approximation formula to each pair of adjacent function values and the corresponding width of the subinterval. The formula for the trapezoidal approximation of a single subinterval is: [ \text{Trapezoid Area} = \frac{\Delta x}{2} \left( f(x_i) + f(x_{i+1}) \right) ] 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 endpoints.

Sum up all the areas of the trapezoids obtained in step 3 to find the total approximate area under the curve.

Calculate the approximate area using the obtained values.
By following these steps, you can find the area using the trapezoidal approximation method for the function ( \cos(4x) ) on the interval ([1, 2]) with ( n = 10 ).
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