Can someone explain to me how to do these?
Given the following definite integral and n=4 answer the following questions. (Round your answers to six decimal places.)
1. Use the Trapezoidal Rule to approximate the definite integral.
Answer: T4=
2. Use the Midpoint Rule to approximate the definite integral.
Answer: M4=
3. Use Simpson's Rule to approximate the definite integral.
Answer: S4=
To answer these questions, we need to first understand the Trapezoidal Rule, Midpoint Rule, and Simpson's Rule for approximating definite integrals.
1. Trapezoidal Rule:
The Trapezoidal Rule approximates the area under a curve by dividing it into trapezoids. To use the Trapezoidal Rule, follow these steps:
- Divide the interval [a, b] into n equal subintervals of width h = (b - a)/n.
- Calculate the function values at the endpoints and the midpoints of each subinterval.
- Sum up the areas of the trapezoids formed by connecting the function values at the endpoints and the midpoints.
For example, if we have the definite integral:
∫[a, b] f(x) dx
The Trapezoidal Rule approximation with n subintervals can be calculated using the formula:
Tn = (h/2) * [f(a) + 2*f(x1) + 2*f(x2) + ... + f(b)]
2. Midpoint Rule:
The Midpoint Rule approximates the area under a curve by dividing it into rectangles. To use the Midpoint Rule, follow these steps:
- Divide the interval [a, b] into n equal subintervals of width h = (b - a)/n.
- Calculate the function values at the midpoints of each subinterval.
- Sum up the areas of the rectangles formed by the function values at the midpoints.
For example, if we have the definite integral:
∫[a, b] f(x) dx
The Midpoint Rule approximation with n subintervals can be calculated using the formula:
Mn = h * [f((a + x1)/2) + f((x1 + x2)/2) + ... + f((bn-1 + bn)/2)]
3. Simpson's Rule:
Simpson's Rule approximates the area under a curve by fitting parabolic arcs to the curve. To use Simpson's Rule, follow these steps:
- Divide the interval [a, b] into n equal subintervals of width h = (b - a)/n.
- Calculate the function values at the endpoints, midpoints, and quarter points of each subinterval.
- Sum up the areas of the parabolic arcs formed by connecting the function values.
For example, if we have the definite integral:
∫[a, b] f(x) dx
The Simpson's Rule approximation with n subintervals can be calculated using the formula:
Sn = (h/3) * [f(a) + 4*f(x1) + 2*f(x2) + 4*f(x3) + ... + 2*f(xn-2) + 4*f(xn-1) + f(b)]
Now, let's apply these rules to the given definite integral with n=4.
1. Using the Trapezoidal Rule:
T4 = (b - a)/2 * [f(a) + 2*f(x1) + 2*f(x2) + 2*f(x3) + f(b)]
2. Using the Midpoint Rule:
M4 = h * [f((a + x1)/2) + f((x1 + x2)/2) + f((x2 + x3)/2) + f((x3 + x4)/2)]
3. Using Simpson's Rule:
S4 = (h/3) * [f(a) + 4*f(x1) + 2*f(x2) + 4*f(x3) + f(b)]
Plug in the given function f(x), the limits of integration [a, b], and calculate the corresponding values using the formulas mentioned above. Round your answers to six decimal places to get the final approximations.
Come on, guy. We just went through this with Riemann sums and trapezoid rule. Just plug in your favorite formula and plug away.
It would also help if you included the actual function involved.