Question

How do you find the area using the trapezoidal approximation method, given `cos(4 x) dx`, on the interval [-1, 2] with n=10?

Answer 1

`int_(-1)^2 \ cos4x \ dx ~~ 0.0510` 4dp

Explanation:

We have:

`y = cos(4x)`

We want to estimate `int \ y \ dx` over the interval `[-1,2]` with `n=10` strips; thus:

`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(sin8-sin(-4)) \ \ \ \` (radians!)
`" \ \ \ = 1/4(0.989358... - 0.756802 ...)`
`" \ \ \ = 0.058138 ...`