Question

(4.2) Consider the integral -f 1 J dec 1+3 (a) Show that (4) 1 da 1 +x3 dr 1+r3 (b) Deduce that (3) -re) J f() dr where f is a function to be determined (8) and the (c) Approximate J by means of the three-term Gaussian Quadrature Hint: The roots of the third Legendre polynomial are xo corresponding coefficients for the three-term Gaussian Quadrature are co =,C= , C2= 15 15, 1 0, 32 5 Y 9 (2) (25] (d) Why is the result in (4 2(c)) not exact? TOTAL MARKS. [100 TUTTCA 2018

Answer 1

SOLUTION- (A)

1 da 13 J =

Consider

1 da 13 I =

Substituting so that

dy y2 dr

1 1 dy y2 I = 1 X

Which is

dy 1+y I =

So that

1 dx da 1

SOLUTION-(C)

Note that

1 dx 1 3 1 da 1+3 1 da 1 0

Which is

That is,
1 da J .2 1
so
f (x) 1 r2
is the function

Using Gaussian quadrature, we have is the approximate integral which equals

SOLUTION-(D)

The result is not exact because is not a polynomial function of degree 2 or less

f (x) 1 r2