Use the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.)
The integral of e^(3sqrt(t)) sin3t dt from 0 to 4. n=8
we want to approximate
∫[0,4] e^(3√t) sin 3t dt
So, divide the interval[0,4] into 8 pieces, each of width 0.5
Now apply each Rule named above, calculating f(x) for each value from 0 to 4, and add up the areas of all the rectangles or trapezoids.
There are online calculators for each of the Rules.
To approximate the given integral using the Trapezoidal Rule, Midpoint Rule, and Simpson's Rule, we need to divide the interval [0,4] into smaller subintervals and calculate the area under the curve within each subinterval.
First, let's find the width of each subinterval using the given value of n. The width, denoted as h, is given by:
h = (b - a) / n,
where a and b are the limits of integration (0 and 4, respectively). Thus, h = (4 - 0) / 8 = 0.5.
Now, let's calculate the approximate values of the integral using each rule:
1. Trapezoidal Rule:
The approximate value of the integral using the Trapezoidal Rule is given by the formula:
∫(a to b) f(x) dx ≈ (h / 2) * [f(a) + 2 * (f(a + h) + f(a + 2h) + ... + f(a + (n-1)h)) + f(b)].
In this case, f(x) = e^(3sqrt(t)) sin(3t).
Using n = 8, we have the following subintervals: [0, 0.5], [0.5, 1], [1, 1.5], [1.5, 2], [2, 2.5], [2.5, 3], [3, 3.5], [3.5, 4].
Substituting the values into the formula, we get:
∫(0 to 4) e^(3sqrt(t)) sin(3t) dt ≈ (0.5 / 2) * [f(0) + 2 * (f(0.5) + f(1) + f(1.5) + f(2) + f(2.5) + f(3) + f(3.5)) + f(4)].
Now, compute the values of f(x) at each interval and substitute them into the formula to get the approximate value.
2. Midpoint Rule:
The approximate value of the integral using the Midpoint Rule is given by the formula:
∫(a to b) f(x) dx ≈ h * [f(a + h/2) + f(a + (3h/2)) + f(a + (5h/2)) + ... + f(b - h/2)].
In this case, f(x) = e^(3sqrt(t)) sin(3t).
Using n = 8, we have the same subintervals: [0, 0.5], [0.5, 1], [1, 1.5], [1.5, 2], [2, 2.5], [2.5, 3], [3, 3.5], [3.5, 4].
Substituting the values into the formula, we get:
∫(0 to 4) e^(3sqrt(t)) sin(3t) dt ≈ 0.5 * [f(0.25) + f(0.75) + f(1.25) + f(1.75) + f(2.25) + f(2.75) + f(3.25) + f(3.75)].
Now, compute the values of f(x) at each midpoint and substitute them into the formula to get the approximate value.
3. Simpson's Rule:
The approximate value of the integral using Simpson's Rule is given by the formula:
∫(a to b) f(x) dx ≈ (h / 3) * [f(a) + 4 * (f(a + h) + f(a + 3h) + ... + f(b - h)) + 2 * (f(a + 2h) + f(a + 4h) + ... + f(b - 2h)) + f(b)].
In this case, f(x) = e^(3sqrt(t)) sin(3t).
Using n = 8, we have the same subintervals: [0, 0.5], [0.5, 1], [1, 1.5], [1.5, 2], [2, 2.5], [2.5, 3], [3, 3.5], [3.5, 4].
Substituting the values into the formula, we get:
∫(0 to 4) e^(3sqrt(t)) sin(3t) dt ≈ (0.5 / 3) * [f(0) + 4 * (f(0.5) + f(1.5) + f(2.5) + f(3.5)) + 2 * (f(1) + f(2) + f(3)) + f(4)].
Now, compute the values of f(x) at each interval and substitute them into the formula to get the approximate value.
Remember to round your answers to six decimal places.