2.2. Use numerical integration to approximate o So ysinxdydx over a rectangular regionwith subintervals m = n = 2 and then determine the percentage deviationbetween the approximated value and the analytic value of the integral.(Calculations should be correct to four decimal places).

Given integral is ∫01∫01ysinxdydx (1) Here f(x, y)=ysinx So f(xi, yj)=yjsinxi Since…

Answered: 2.2. Use numerical integration to…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

2.2. Use numerical integration to approximate o So ysinxdydx over a rectangular region
with subintervals m = n = 2 and then determine the percentage deviation
between the approximated value and the analytic value of the integral.
(Calculations should be correct to four decimal places).

Expert Solution

Step 1

Given integral is

$\int_{0}^{1} \int_{0}^{1} y \sin x d y d x$ (1)

Here $f (x, y) = y \sin x$

So $f (x_{i}, y_{j}) = y_{j} \sin x_{i}$

Since there is two subintervals, step size will be $h = k = 0.5$

So table is

$y$ $x$	0.5	1
0	0	0
0.5	0.2397	0.4207
1	0.4794	0..8415

Formula of trapezoidal rule for double integration is

$\int_{x_{0}}^{x_{2}} \int_{y_{0}}^{y_{2}} f (x, y) d y d x = \frac{h k}{4} [f (x_{0}, y_{0}) + 2 f (x_{0}, y_{1}) + f (x_{0}, y_{2}) + 2 f (x_{1}, y_{0}) + 4 f (x_{1}, y_{1}) + 2 f (x_{1}, y_{2}) + f (x_{2}, y_{0}) + 2 f (x_{2}, y_{1}) + f (x_{2}, y_{2})]$