Help me with trapezoidal approximation?

Question

The trapezoidal approximation of ∫ sin x dx from 0 to pi using 4 equal subdivisions of the interval of integration is 

 

a) pi/2

b) pi

c) (pi/4)(1+sqrt(2))

d) (pi/2)(1+sqrt(2))

e) (pi/4)(2+sqrt(2))

 

Answer: C

 

Please show all your work.

Answer 1

The trapezoidal approximation for the definite integral of f(x) between the points a and b is:

 

T_n = [(b-a)/2n][f(x₀)+2f(x₁)+2f(x₂)+2f(x₃)+...+f(x_n)]

 

Where a and b are the limits of the integration and n = the number of trapezoids used in the estimate.  In your case:

 

f(x)= sinx

a=x₀=0, b=x_n=pi, n=4

 

T₄ = (pi/8)[sin(0)+2sin(pi/4)+2sin(pi/2)+2sin(3pi/2)+sin(pi)]

 

Solve for T₄