a=3t2
(i) v=∫a dt=∫3t2 dt=t3
v(0)=03=0m/sv(2.2)=2.23=10.648m/s
(ii) ∫02.23t2 dt
f(x)=3t2,a=0,b=2.2
The width of each subinterval is
Δx=nb−a=62.2=0.37
so the grid points have the coordinates
xi=i0.37
Calculate the values of the function f(x) at the points xi:
f(x0)=f(0)=3(02)=02=0
f(x1)=f(0.37)=3(0.372)=0.4107
f(x2)=f(2(0.37))=3(0.742)=1.6428
f(x3)=f(3(0.37))=3(1.112)=3.6963
f(x4)=f(4(0.37))=3(1.482)=6.5712
f(x5)=f(6(0.37))=3(1.852)=10.2675
f(x6)=f(2(0.37))=3(2.222)=14.7852
The Trapezoidal Rule formula is written in the form;
∫02.23t2 dt≈T6=2Δx[f(x0)+2f(x1)+⋯+2f(x5)+f(x6)]=20.37[0+2⋅0.4107+2⋅1.6428+2⋅3.6963+2⋅6.5712+2⋅10.2675+2⋅14.7852]=
19.2986 m/s
(iii) ∫02.2a dt=∫02.23t2 dt=[t3]02.2=2.23−03=10.648 m/s
which coincides with the original value
(iv) all numerical steps have affected the original obtained result.
Comments