We are tasked with estimating the integral $\int_0^1 12 \cos(x^2) \, dx$ using the Trapezoidal Rule and the Midpoint Rule with $n = 4$.

Step 1: Subdivide the Interval

The interval $[0, 1]$ is divided into $n = 4$ subintervals, so the width of each subinterval is: $\Delta x = \frac{1 - 0}{4} = 0.25$

The points where we will evaluate the function are:

• Trapezoidal Rule: $x_0 = 0$, $x_1 = 0.25$, $x_2 = 0.5$, $x_3 = 0.75$, $x_4 = 1$
• Midpoint Rule: The midpoints of the intervals: $x_{mid,1} = 0.125$, $x_{mid,2} = 0.375$, $x_{mid,3} = 0.625$, $x_{mid,4} = 0.875$

Step 2: Apply Trapezoidal Rule

The Trapezoidal Rule formula is: $T = \frac{\Delta x}{2} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + f(x_4) \right]$ where $f(x) = 12 \cos(x^2)$.

Step 3: Apply Midpoint Rule

The Midpoint Rule formula is: $M = \Delta x \left[ f(x_{mid,1}) + f(x_{mid,2}) + f(x_{mid,3}) + f(x_{mid,4}) \right]$

Step 4: Numerical Computations

I will now calculate these values numerically. Let me compute them.The numerical estimates for the integral $\int_0^1 12 \cos(x^2) \, dx$ using $n = 4$ are as follows:

• Trapezoidal Rule Estimate: $T \approx 10.749$
• Midpoint Rule Estimate: $M \approx 10.907$

Step 5: Analyze Overestimate or Underestimate

From the graph of the function $12 \cos(x^2)$, we observe that the function is concave down on the interval $[0, 1]$ because its second derivative $\frac{d^2}{dx^2}(12 \cos(x^2))$ is negative in this region. This means:

• The Trapezoidal Rule tends to underestimate the integral because it under-approximates areas under concave down curves.
• The Midpoint Rule tends to overestimate the integral because it over-approximates the area for concave down curves.

Conclusion:

Since the Trapezoidal Rule provides an underestimate and the Midpoint Rule provides an overestimate, the true value of the integral lies between these two estimates. Therefore, we can conclude: $10.749 < \int_0^1 12 \cos(x^2) \, dx < 10.907$

Would you like more details on any part of the calculation?

Here are 5 related questions you might find useful:

1. How do you compute the error bounds for the Trapezoidal Rule and Midpoint Rule?
2. How would increasing $n$ affect the accuracy of these estimates?
3. Can Simpson’s Rule be applied to this problem for a more accurate estimate?
4. What is the exact value of this integral, and how does it compare with these estimates?
5. How does the concavity of a function affect the accuracy of different numerical integration methods?

Tip: Always graph the function before applying numerical integration techniques to better understand the behavior of the curve and the expected accuracy of the methods.