evaluate the following integral : integral from 0 to 4 of (2-e^(-3x))dx and solve the following b. the trapezoidal rule with n=4 in matlab

To evaluate the integral ∫(0 to 4) (2 - e^(-3x)) dx, you can use antiderivative method or numerical integration. Here, I will show you how to solve it using the antiderivative method:

First, let's find the antiderivative of the function: ∫(2 - e^(-3x)) dx = 2x - (1/3)e^(-3x)

Now, let's evaluate the definite integral by substituting the upper and lower limits: ∫(0 to 4) (2 - e^(-3x)) dx = [2x - (1/3)e^(-3x)] from 0 to 4 = [2(4) - (1/3)e^(-3(4))] - [2(0) - (1/3)e^(-3(0))] = 8 - (1/3)e^(-12) - 0 + (1/3)e^0 = 8 - (1/3)e^(-12) + (1/3)

Therefore, the value of the integral ∫(0 to 4) (2 - e^(-3x)) dx is 8 - (1/3)e^(-12) + (1/3).

Now, let's solve b. the trapezoidal rule with n=4 in MATLAB to approximate the integral.

The trapezoidal rule is a numerical method for approximating definite integrals. The formula for the trapezoidal rule with n subintervals is:

∫(a to b) f(x) dx ≈ h/2 * [f(a) + 2f(a+h) + 2f(a+2h) + ... + f(b)]

Where h = (b - a) / n is the width of each subinterval.

In MATLAB, you can implement the trapezoidal rule as follows:

main.m
a = 0;  % Lower limit of integration
b = 4;  % Upper limit of integration
n = 4;  % Number of subintervals
h = (b - a) / n;  % Width of each subinterval

% Define the function f(x)
f = @(x) 2 - exp(-3*x);

% Calculate the approximation using the trapezoidal rule
approximation = (h/2) * (f(a) + 2*sum(f(a + (1:n)*h)) + f(b));

% Display the result
disp(approximation);
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Running this MATLAB code will give you the approximation of the integral using the trapezoidal rule with n=4.