5. -3 points My Notes SEssCalcET1 6.5.008 to approximate the given integral with

Question

5. -3 points My Notes SEssCalcET1 6.5.008 to approximate the given integral with the specifi ed value of n. (Round your answers to six Use the Trapezoidal Rule, the Midpoint Rule and Simpson's Rule decima places.) 1/2 6 sin x2 dx, n 4 (a) the Trapezoidal Rule (b) the Midpoint Rule (c) Simpson's Rule Need Help? Read it Chat About It

Explanation / Answer

int 6 sin(x^2) dx, n=4 a) trapezoidal rule f(x) = 6 sin(x^2) a= 0 b=1/2=0.5 n=4 h=stepsize = (b-a)/n = (0.5-0)/4 = 0.125 x0=a x1= x0+h = a+h = 0+0.125 x2= x0+2 h = a+2h = 0+ 2*(0.125 ) x3= x0+3 h = a+3h = 0+ 3*(0.125 ) x4= x0+4 h = a+4h =0+ 4*(0.125 ) int 6 sin(x^2) dx = h ( 0.5 f(x0) + f(x1) + f(x2 ) + f(x3 ) + 0.5*f(x 4 ) ) = h ( 0.5 f(a) + f(a+h) + f(a+ 2*h ) + f(a+ 3*h ) + 0.5*f(a+ 4*h ) ) = 0.125* ( 0.5 f(0) + f(0+0.125) + f(0+ 2*(0.125) ) + f(0+ 3*(0.125) ) + 0.5*f(0+ 4*(0.125) ) ) = 0.125* ( 0 + 0.0937462 +0.374756 + 0.840972 + 0.5*(1.48442) ) = 0.256461 b) midpoint rule f(x) = 6 sin(x^2) a= 0 b=1/2=0.5 n=4 h=stepsize = (b-a)/n = (0.5-0)/4 = 0.125 x0=a xibar = a+ (i-1/2) h i=1==> x1bar= x0+(1-1/2)h = 0+ (0.125) /2 = 0.0625 i=2==> x2bar= x0+(2-1/2) h = a+3h/2 = 0+ 3*(0.125 )/2 = 0.1875 i=3==> x3bar= x0+(3-1/2) h = a+5h/2 = 0+ 5*(0.125 )/2 = 0.3125 i=4==> x4bar= x0+(4-1/2) h = a+7h/2 =0+ 7*(0.125 )/2 = 0.4375 int 6 sin(x^2) dx = h ( f(x1bar) + f(x2bar ) + f(x3bar ) +f(x4bar ) ) = 0.125* ( f(0.0625) + f( 0.1875 ) + f(0.3125 ) +f(0.4375 ) ) = 0.125* ( 0.0234374 + 0.210894 + 0.585007 + 1.14144 ) = 0.245097 c) Simpson's rule f(x) = 6 sin(x^2) a= 0 b=1/2=0.5 n=4 h=stepsize = (b-a)/n = (0.5-0)/4 = 0.125 x0=a x1= x0+h = a+h = 0+0.125 x2= x0+2 h = a+2h = 0+ 2*(0.125 ) x3= x0+3 h = a+3h = 0+ 3*(0.125 ) x4= x0+4 h = a+4h =0+ 4*(0.125 ) int 6 sin(x^2) dx = (h/3) ( f(x0) + 4* f(x1) + 2*f(x2 ) + 4*f(x3 ) + f(x 4 ) ) = (h/3) ( f(a) +4 f(a+h) +2 f(a+ 2*h ) + 4 f(a+ 3*h ) + f(a+ 4*h ) ) = (0.125/3) * ( f(0) + 4 f(0+0.125) +2 f(0+ 2*(0.125) ) + 4 f(0+ 3*(0.125) ) + f(0+ 4*(0.125) ) ) = (0.125/3) * ( 0 + 4* 0.0937462 +2*0.374756 +4* 0.840972 + 1.48442 ) = 0.248867

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