Given f(x) > 0 with f ′(x) > 0, and f ′′(x) > 0 for all x in the interval [0, 3] with f(0) = 0.1 and f(3) = 1, the left, right, trapezoidal, and midpoint rule approximations were used to estimate the integral from 0 to 3 of f of x, dx
f is positive, increasing, and concave up.
Draw such a curve, and you can see which approximations over- and under-estimate the value.
To approximate the integral from 0 to 3 of f(x) dx using the left, right, trapezoidal, and midpoint rule approximations, follow these steps:
1. Divide the interval [0, 3] into n equal subintervals of width h = (b - a) / n, where a = 0 and b = 3.
2. Calculate the value of h using the given interval [0, 3] and the number of subintervals n. Choose an appropriate value for n to ensure accuracy.
3. Evaluate f(x) at the left endpoints of the subintervals. This will give you the left sum approximation.
4. Evaluate f(x) at the right endpoints of the subintervals. This will give you the right sum approximation.
5. Evaluate f(x) at the midpoints of the subintervals. This will give you the midpoint rule approximation.
6. Calculate the area of each trapezoid by averaging the values of f(x) at the left and right endpoints of each subinterval. Sum up these areas to obtain the trapezoidal rule approximation.
7. Finally, calculate the weighted average of the left and right sum approximations to obtain the composite midpoint rule approximation.
Note: The formulas for the approximations are as follows:
Left sum approximation:
L_n = h * (f(x_0) + f(x_1) + f(x_2) + ... + f(x_{n-1}))
Right sum approximation:
R_n = h * (f(x_1) + f(x_2) + f(x_3) + ... + f(x_n))
Midpoint rule approximation:
M_n = h * (f((x_0 + x_1) / 2) + f((x_1 + x_2) / 2) + f((x_2 + x_3) / 2) + ... + f((x_{n-1} + x_n) / 2))
Trapezoidal rule approximation:
T_n = (h/2) * (f(x_0) + 2f(x_1) + 2f(x_2) + ... + 2f(x_{n-1}) + f(x_n))
Composite midpoint rule approximation:
C_n = (L_n + R_n) / 2
where x_0, x_1, x_2, ..., x_n are the endpoints of each subinterval.
Evaluate each of the approximations using the given values to obtain numerical estimates for the integral of f(x) from 0 to 3.
To estimate the integral from 0 to 3 of f(x) dx using different numerical methods, we can use the left rule, right rule, trapezoidal rule, and midpoint rule.
1. Left Rule:
The left rule approximates the integral by using the value of the function at the left endpoints of each subinterval. The formula for the left rule is:
L = h * (f(x0) + f(x1) + ... + f(xn-1))
where h is the width of each subinterval and n is the number of subintervals.
2. Right Rule:
The right rule approximates the integral by using the value of the function at the right endpoints of each subinterval. The formula for the right rule is:
R = h * (f(x1) + f(x2) + ... + f(xn))
where h is the width of each subinterval and n is the number of subintervals.
3. Trapezoidal Rule:
The trapezoidal rule approximates the integral by using the average of the values of the function at the left and right endpoints of each subinterval. The formula for the trapezoidal rule is:
T = (h/2) * (f(x0) + 2f(x1) + 2f(x2) + ... + 2f(xn-1) + f(xn))
where h is the width of each subinterval and n is the number of subintervals.
4. Midpoint Rule:
The midpoint rule approximates the integral by using the value of the function at the midpoint of each subinterval. The formula for the midpoint rule is:
M = h * (f((x0 + x1)/2) + f((x1 + x2)/2) + ... + f((xn-1 + xn)/2))
where h is the width of each subinterval and n is the number of subintervals.
To find the values of f(x) at each endpoint or midpoint, we evaluate f(x) with the given conditions: f(0) = 0.1 and f(3) = 1.
Once we have the values of f(x) at each endpoint or midpoint, we can calculate the approximations using the formulas provided for each method.
Remember to divide the interval [0, 3] into smaller subintervals to get a more accurate estimation. The larger the number of subintervals, the more accurate the approximation will be.