How do you find the area using the trapezoidal approximation method, given #f(x)=5 sqrt(1+sqrt(x)) #, on the interval [0, 4] with n=8?

1 Answer
Jul 7, 2017

Answer:

# int_0^4 \ 5sqrt(1+sqrt(x)) \ dx ~~ 30.2149#

Explanation:

We have:

# f(x) = 5sqrt(1+sqrt(x)) #

We want to calculate over the interval #[0,4]# with #8# strips; thus:

# Deltax = (4-0)/8 = 1/2#

Note that we have a fixed interval (strictly speaking a Riemann sum can have a varying sized partition width). The values of the function are tabulated as follows;

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Trapezium Rule

# int_0^4 \ 5sqrt(1+sqrt(x)) \ dx ~~ 0.5/2 * { 5 + 8.6603 + #
# " " 2*(6.5328 + 7.0711 + 7.4578 + #
# " " 7.7689 + 8.033 + 8.2645 + 8.4718) } #

# " " = 0.25 * { 13.6603 + 2*(53.5997) }#
# " " = 0.25 * { 13.6603 + 107.1994 }#
# " " = 0.25 * 120.8597#
# " " = 30.2149#

Actual Value

For comparison of accuracy:

# int_0^4 \ 5sqrt(1+sqrt(x)) \ dx = 30.3794795877687... #