f(x)=2x+3; on [0,4] with 4 subintervals

To find the integral of the function f(x) = 2x + 3 over the interval [0, 4] with 4 subintervals, you can use the numerical integration technique called the Trapezoidal Rule.

Step 1: Divide the interval [0, 4] into 4 equal subintervals.
The interval [0, 4] is divided into 4 subintervals of equal width by using the formula:
Δx = (b - a) / n
Where a = 0 (the lower limit of the interval), b = 4 (the upper limit of the interval), and n = 4 (the number of subintervals).
Therefore, Δx = (4 - 0) / 4 = 1.

The subintervals are:
[0, 1], [1, 2], [2, 3], [3, 4]

Step 2: Evaluate the function at the endpoints of the subintervals.
For each subinterval, substitute the endpoints into the function f(x) = 2x + 3.
For the first subinterval [0, 1]:
f(0) = 2(0) + 3 = 3
f(1) = 2(1) + 3 = 5

For the second subinterval [1, 2]:
f(1) = 2(1) + 3 = 5
f(2) = 2(2) + 3 = 7

For the third subinterval [2, 3]:
f(2) = 2(2) + 3 = 7
f(3) = 2(3) + 3 = 9

For the fourth subinterval [3, 4]:
f(3) = 2(3) + 3 = 9
f(4) = 2(4) + 3 = 11

Step 3: Calculate the sum of the function evaluations multiplied by the width of the subintervals.
Apply the Trapezoidal Rule formula:
∫[a, b] f(x) dx ≈ Δx/2 * [f(a) + 2f(x1) + 2f(x2) + ... + f(b)]
Where Δx is the width of each subinterval, f(a) and f(b) are the function evaluations at the first and last endpoints, and f(xi) are the function evaluations at the remaining interior endpoints of the subintervals.

Apply the formula for our case:
∫[0, 4] (2x + 3) dx ≈ (1/2) * [3 + 2(5) + 2(7) + 2(9) + 11]
≈ (1/2) * [3 + 10 + 14 + 18 + 11]
≈ (1/2) * 56
≈ 28

Therefore, the approximate value of the integral ∫[0, 4] (2x + 3) dx using the Trapezoidal Rule with 4 subintervals is 28.