How do you use the Trapezoidal Rule with step size n=4 to estimate #int t^3 +t) dx# with [0,2]?

Answer 1

Assuming that the variables should match see below.

Approximate the Integral #int_a^b f(x) dx# using trapezoidal approximation with #n# intervals.
In this question we have: #f(x) = x^3+x# #{a,b] = [0, 2]#, and #n=4#.
So we get #Delta x = (b-a)/n = (2-0)/4 = 1/2#
The endpoints of the subintervals are found by beginning at #a=0# and successively adding #Delta x = 1/2# to find the points until we get to #x_n = b = 4#.
#x_0 = 0#, #x_1 = 1/2#, #x_2 =2/2 = 1#, #x_3 = 3/2# and #x_4 = 4/2 = 2 = b#
Now apply the formula (do the arithmetic) for #f(x) = x^3+x#.
#T_4=1/2Deltax [f(x_0)+2f(x_1)+2f(x_2)+2f(x_3) + f(x_4)] #
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To use the Trapezoidal Rule with step size ( n = 4 ) to estimate ( \int_{0}^{2} (t^3 + t) , dt ), follow these steps:

  1. Divide the interval ( [0, 2] ) into ( n = 4 ) equal subintervals. Each subinterval will have a width of ( \Delta x = \frac{2 - 0}{4} = 0.5 ).
  2. Calculate the function values at the endpoints of each subinterval: ( f(0), f(0.5), f(1), f(1.5), f(2) ), where ( f(t) = t^3 + t ).
  3. Apply the Trapezoidal Rule formula: [ \int_{a}^{b} f(x) , dx \approx \frac{\Delta x}{2} [f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + f(x_4)] ] where ( \Delta x = 0.5 ), and ( x_0 = 0, x_1 = 0.5, x_2 = 1, x_3 = 1.5, x_4 = 2 ).
  4. Substitute the values into the formula and calculate the approximation.
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7