Question

Find the approximations T10, M10, and S10 for π 7 sin x dx. 0 (Round your answers to six decimal places.)

Answer 1

Explanation:

Well, since it was not given the interval let's use the interval [0,5] with n=10

So now, for the Trapezoidal Rule to approximate the area enclosed by the Integral of:
`f(x)=7\pi \sin(x)`

`T_(10)=(b-a)/(2n)[f(a)+2f(x_1)+ ....2f(x_(n-1))+f(b)]` Plugging in:

`T_(10)=(5-0)/(2*10)[f(0)+2f((1)/(2))+2f(1)+2f((3)/(2))+2f(2)+2f(5/2)+2f(3)+2f(7/2)+2f(4)+2f(9/2) +f(5)]`

`T_(10)=(1)/(4)[0+21.086+37+43.87+39.99+26.322+6.20-15.43-33.285-42.99-21.087]`

`T_(10)\approx 15.419`

Now the same area according to Simpson rule:

`S_(10)=(b-a)/(3n)[f(a)+4f(x_(1))+2f(x_(2))+4f(x_(3) )+2f(x_(4))+4f(x_(5))+2f(x_(6))+4f(x_(7))+2f(x_(8))+4f(x_(9))+f(b)]\\S_(10)=(5)/(3*10)[0+74.01+43.87+79.98+26.322+12.413-15.43-66.571-42.99-21.08]\approx 15.085`

`S_(10)\approx 15.0585`