# Use the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule to approximate the given integral...

## Question:

Use the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule to approximate the given integral with the specified value of n.

{eq}e^4 \sqrt{t sin 6t} {/eq} dt , n = 8 from 0 to 4.

## Integral Approximation:

To solve this problem we will use the following theorems:

1.) Midpoint Rule:

{eq}\int_a^{b} f(x) \, dx \approx \, M_n = \Delta x \left[ f(\bar x_1) + f(\bar x_2) + ......+ f(\bar x_n) \right] {/eq}

Where,

{eq}\Delta x = \frac{b- a}{n} {/eq}

And

{eq}\bar x_i = \frac{1}{2} (x_{(i-1)} + x_i) = \, \, midpoint \, of \, \, \left[ x_{(i-1)},x_i \right] {/eq}

2.) Trapezoidal Rule:

{eq}\int_a^{b} f(x) \, dx \approx T_n = \frac{\Delta x}{2} \left[ f(x_0) + 2f(x_1) + ......+ 2f(x_{(n-1)}) + f(x_n) \right] {/eq}

Where,

{eq}\Delta x = \frac{b- a}{n} \, \, and \, \, x_i = a + i \, \Delta x {/eq}

3.) Simpson's Rule:

{eq}\int_a^{b} f(x) \, dx \approx S_n = \frac{\Delta x}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2)+......+2f(x_{(n-2)})+ 4f(x_{(n-1)}) + f(x_n) \right] {/eq}

Where,

{eq}\Delta x = \frac{b- a}{n} \, \, and \, \, n {/eq} is even

We are given:

{eq}n = 8 \, , \, \Delta x = \frac{4- 0}{8}= 0.5 \, ,\, a = 0 \, and \, b = 4 {/eq}

Let us assume that {eq}f(t) = e^{4} \sqrt{t...

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