Question

How do you find the area using the trapezoidal approximation method, given `e^(x^2)`, on the interval [0,1] with n=10?

Answer 1

`int_0^1 e^(x^2) \ dx ~~ 1.4672` (4dp)

Explanation:

The values of `f(x)=e^(x^2)` are tabulated as follows (using Excel) working to 6dp.

The Trapezium Rule:

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

uses a series of two consecutive ordinates and a best fit straight line to form trapeziums to approximate the area under a curve, It will have 100% accuracy if `y=f(x)` is a straight line and typically provides a good estimate provided `n` is chosen appropriately.

So,

`int_0^1 e^(x^2) \ dx ~~ 0.1/2 { (1 + 2.718281) +`
`" " 2(1.01005 + 1.04081 + 1.094174 + 1.17351 + 1.284025 +`
`" " 1.433329 + 1.632316 + 1.89648 + 2.247907) }`
`" "= 0.05 { + 3.718281 + 2(12.812606) }`
`" "= 0.05 { + 3.718281 + 25.625212 }`
`" "= 0.05 { + 29.343493 }`
`" "= 1.467174`