Question

How do you use the trapezoidal rule with n = 4 to approximate the area bounded by the curves `y = sin2x`, the lines x = 0, y = 0, and x = 1?

Answer 1

`int_0^1sin(2x)dx ~~ 0.693` (3dp)

Explanation:

The values of y are tabulated as follows (using Excel)

Using the trapezoidal rule:
`int_a^bydx ~~ h/2{(y_0+y_n)+2(y_1+y_2+...+y_(n-1))}`

we have:
`int_0^1sin(2x)dx ~~ 0.25/2{ (0.00000 + 0.90930 + 2(0.47943 + 0.84147 + 0.99749)}`

`:. int_0^1sin(2x)dx ~~ 0.125{ 0.90930 + 2(2.31839)}`
`:. int_0^1sin(2x)dx ~~ 0.125{ 0.90930 + 4.63678}`
`:. int_0^1sin(2x)dx ~~ 0.125{ 5.54608}`
`:. int_0^1sin(2x)dx ~~ 0.69326`