How do you use the trapezoidal rule with n=4 to approximate the area between the curve # (5t + 6)# from 3 to 6?

1 Answer
Jan 3, 2017

Answer:

# int_(3)^(6) 5t+6 \ dt =85.5 #

Explanation:

The required area is a trapezium with sides #21# and #36# and width #3#, so we can calculate the exact area as #A=1/2(21+36)3 = 85.5#. As the trapezium rule estimates the area under a curve using trapeziums we would expect to get an exact answer.

The values of #f(x)=5x+6# are tabulated as follows (using Excel) working to 5dp

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Using the trapezoidal rule:

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

We have:

# int_(3)^(6) 5t+6 \ dt ~~ 0.75/2 { 21 + 36 + 2(24.75 + 28.5 + 32.25)} #
# " " = 0.375 { 57 + 2( 85.5 )}#
# " " = 0.375 { 57 + 171 }#
# " " = 0.375 { 228 }#
# " " = 85.5 " "#, as predicted