a.) calculate the approximation for Trapezoidal and midpoint rules when n=4 for

Question

a.) calculate the approximation for Trapezoidal and midpoint rules when n=4 for the integral 5e^(1/x) from 1 to 2. ROUND TO 6 DECIMAL PLACES.

b.) estimate the errors of the approximations of part a). ROUND TO 6 DECIMALS.

c.) How large does n have to be so that the answers in part a.) are .00001 off.

Can someone give me the stp by step too please?

2. How large should n be so that by the simpsons rule approx. the integral of 19e^x^2 from 0 to 1 is accurate to within .00001

WHOEVER DOES THIS WILL BE A GIGANTIC HELP INTO A YOUNG WOMANS PERSUIT IN ENGINEERING :)

Explanation / Answer

(a) calculate the approximation for Trapezoidal and midpoint rules when n=4 for the integral 5e^(1/x) from 1 to 2.

Height Value:

delta(x) = (2 - 1)/4
delta(x) = 1/4
delta(x) = 0.25

Integral using Trapezoidal Rule

= (1/2) * (0.25) * ( f(1) + 2*f(1.25) + 2*f(1.5) + 2*f(1.75) + f(2) )
= (1/2) * (0.25) * ( 13.591409142295 + 2*11.127704642462 + 2*9.7386702052734 + 2*8.8539747621758 + 8.2436063535006 )
= 10.159464339452

Integral using Midpoint Rule

= (0.25) * ( f((1+1.25)/2)) + f((1.25+1.5)/2)) + f((1.5+1.75)/2)) + f((1.75+2)/2)) )
= (0.25) * ( f(1.125) + f(1.375) + f(1.625) + f(1.875) )
= (0.25) * ( 12.162127271436 + 10.347145035785 + 9.2518407138462 + 8.5230243266138 )
= 10.07103433692

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