Calculating Definite Integrals: Trapezoid, Simpson's, Exact Value

Hey guys! Today, we're diving deep into the world of definite integrals, exploring different methods to solve them. We'll be focusing on the integral $\int_{-4}^{-2} 19 x^2 d x$, and we're going to tackle it using three cool techniques: the Trapezoid Rule, Simpson's Rule, and finding the exact value through direct integration. So, buckle up and let's get started!

a) Approximate the Definite Integral with the Trapezoid Rule and $n = 4$

Alright, let's kick things off with the Trapezoid Rule. This method approximates the area under a curve by dividing it into trapezoids and summing their areas. For the integral $\int_{-4}^{-2} 19 x^2 d x$, we're given $n = 4$, which means we'll be dividing the interval $[-4, -2]$ into four equal subintervals. Let's break it down step by step.

First, we need to find the width of each subinterval, denoted as $\Delta x$. The formula for $\Delta x$ is:

$\Delta x = \frac{b - a}{n}$

where $a$ is the lower limit of integration, $b$ is the upper limit, and $n$ is the number of subintervals. In our case, $a = -4$, $b = -2$, and $n = 4$. Plugging these values in, we get:

$\Delta x = \frac{-2 - (-4)}{4} = \frac{2}{4} = 0.5$

Now that we have $\Delta x$, we can find the endpoints of our subintervals. These endpoints are $x_0, x_1, x_2, x_3,$ and $x_4$. They are:

• $x_0 = -4$
• $x_1 = -4 + 0.5 = -3.5$
• $x_2 = -3.5 + 0.5 = -3$
• $x_3 = -3 + 0.5 = -2.5$
• $x_4 = -2$

Next, we need to evaluate the function $f(x) = 19x^2$ at each of these endpoints. So, we calculate:

• $f(x_0) = f(-4) = 19(-4)^2 = 19(16) = 304$
• $f(x_1) = f(-3.5) = 19(-3.5)^2 = 19(12.25) = 232.75$
• $f(x_2) = f(-3) = 19(-3)^2 = 19(9) = 171$
• $f(x_3) = f(-2.5) = 19(-2.5)^2 = 19(6.25) = 118.75$
• $f(x_4) = f(-2) = 19(-2)^2 = 19(4) = 76$

Now we can apply the Trapezoid Rule formula:

$\int_{a}^{b} f(x) dx \approx \frac{\Delta x}{2} [f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + f(x_4)]$

Plugging in the values we calculated, we get:

$\int_{-4}^{-2} 19x^2 dx \approx \frac{0.5}{2} [304 + 2(232.75) + 2(171) + 2(118.75) + 76]$ $\approx 0.25 [304 + 465.5 + 342 + 237.5 + 76]$ $\approx 0.25 [1425]$ $\approx 356.25$

So, using the Trapezoid Rule with $n = 4$, we approximate the definite integral $\int_{-4}^{-2} 19 x^2 d x$ to be approximately 356.25. Keep in mind, this is just an approximation, and the accuracy depends on the number of subintervals we use. The more subintervals, the better the approximation.

b) Approximate the Definite Integral with Simpson's Rule and $n=4$

Next up, we're going to use Simpson's Rule to approximate the same definite integral. Simpson's Rule is generally more accurate than the Trapezoid Rule because it uses parabolas instead of straight lines to approximate the curve. Again, we have $n = 4$ for the integral $\int_{-4}^{-2} 19 x^2 d x$.

Since $n = 4$, we already have the endpoints and the function values at those endpoints from our Trapezoid Rule calculation. These are:

• $x_0 = -4$, $f(x_0) = 304$
• $x_1 = -3.5$, $f(x_1) = 232.75$
• $x_2 = -3$, $f(x_2) = 171$
• $x_3 = -2.5$, $f(x_3) = 118.75$
• $x_4 = -2$, $f(x_4) = 76$

Also, remember that $\Delta x = 0.5$.

Now, let's apply Simpson's Rule formula:

$\int_{a}^{b} f(x) dx \approx \frac{\Delta x}{3} [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + f(x_4)]$

Plugging in the values, we get:

$\int_{-4}^{-2} 19x^2 dx \approx \frac{0.5}{3} [304 + 4(232.75) + 2(171) + 4(118.75) + 76]$ $\approx \frac{0.5}{3} [304 + 931 + 342 + 475 + 76]$ $\approx \frac{0.5}{3} [2128]$ $\approx \frac{1064}{3}$ $\approx 354.6667$

So, using Simpson's Rule with $n = 4$, we approximate the definite integral $\int_{-4}^{-2} 19 x^2 d x$ to be approximately 354.6667. As expected, this is a more accurate approximation than the Trapezoid Rule. Simpson's Rule provides a better estimate of the area under the curve by using parabolic segments instead of trapezoids, which reduces the error in the approximation.

c) Find the Exact Value of the Integral

Finally, let's find the exact value of the integral using direct integration. This will give us a benchmark to compare the accuracy of the Trapezoid and Simpson's Rules.

We start with the integral:

$\int_{-4}^{-2} 19 x^2 d x$

First, we find the antiderivative of $19x^2$:

$\int 19x^2 dx = 19 \int x^2 dx = 19 \cdot \frac{x^3}{3} + C = \frac{19}{3}x^3 + C$

Now, we evaluate the definite integral by plugging in the limits of integration:

$\int_{-4}^{-2} 19 x^2 d x = \frac{19}{3}[(-2)^3 - (-4)^3]$ $\ = \frac{19}{3}[-8 - (-64)]$ $\ = \frac{19}{3}[-8 + 64]$ $\ = \frac{19}{3}[56]$ $\ = \frac{1064}{3}$ $\approx 354.6667$

So, the exact value of the integral is $\frac{1064}{3}$, which is approximately 354.6667. Notice that Simpson's Rule gave us a very accurate approximation, matching the exact value up to several decimal places. The Trapezoid Rule, while easier to compute, was a bit further off.

In conclusion, we've calculated the definite integral $\int_{-4}^{-2} 19 x^2 d x$ using the Trapezoid Rule, Simpson's Rule, and direct integration. We found that Simpson's Rule provided a much closer approximation to the exact value compared to the Trapezoid Rule. Direct integration gave us the exact value, which served as a great comparison point. Hope you found this helpful, and happy integrating!