Question

QUESTION Numerical integration method 1) Newton-Cotes Rules 2) Gauss Legendre Rules 3) Euler Method 4) Runge-Kutta Method 5) Trapezoidal Method 6) Milne's Method 7) Adams-Moulton-Bashforth By using ONLY ONE METHOD of the aforementioned numerical integration method, determine the estimated surface area of a lake. Given that the maximum length of the lake is 63.49 km, the maximum width of the lake is 22.8 km, the maximum depth of the lake is 104 m, the shore length is 235.2 km and the surface elevation of the lake is 85.6 km. Give your answer in 5 decimal points.

Answer 1

given that the

maximum length of the lake = 63.49km

maximum width of the lake = 22.8km

maximum depth of the lake = 104m

shore length of the lake = 235.2km

surface elevation of the lake = 85.6km

we use the trapezoidal rule to find the estimated surface area of the lake

evaluate a definite integral _a^b∫f(x)dx.

Let f(x) be continuous on [a,b]. We partition the interval [a,b] into n equal subintervals, each of width

Δx=b−a / n,

such that a=x0<x1<x2<⋯<xn=b.

The Trapezoidal Rule for approximating ^b_a∫f(x)dx is given by

_a^b∫f(x)dx≈Tn=Δx2[f(x0)+2f(x1)+2f(x2)+⋯+2f(xn−1)+f(xn)],

where Δx=b−an and xi=a+iΔx.

As n→∞, the right-hand side of the expression approaches the definite integral b∫af(x)dx.

here let us consider length ab as shore length ie 235.2km

divide shore length into 2 it becomes 117.6 km

Tn=117.6/2(0+2(22.8))

= 117.6/2(45.6)

= 5,362.56/2

= 2681.28000 km

this is the estimated surface area of a lake.

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