How do you find the area using the trapezoidal approximation method, given #(5t + 6) dt #, on the interval [3, 6] with n=4?

1 Answer
Steve M
Jan 31, 2017

#85.5 \ "unit"^2#

Explanation:

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

enter image source here

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#

It is always worth thinking about the interpretation of a mathematical question and/or result. In this case we are being asked to find the area under a straight line, which will be a trapezium

graph{5x+6 [-3, 8, -5, 55]}

So we can calculate the exact area:

# A= (1/2)(21+36)(3) = 85.5#

It is no coincidence that the above trapezium rule calculation gave us the exact result, because obviously the rule estimates the area of integration with small width trapeziums. In the case of a straight line those trapeziums are exact.

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