Question

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?

Answer 1

`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;

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...`