give answer in 2 step with explanation at the end of each step and final answer at the solved all parts end:1. Letf(8) = eon We will be estimating the integral Jy f@)do.
a. [2 pts] Use a computer sy

Question

give answer in 2 step with explanation at the end of each step and final answer at the solved all parts  end:1. Letf(8) = eon We will be estimating the integral Jy f@)do.
a. [2 pts] Use a computer system to graph two copies of the graph of f(8). Make sure
your graph includes the interval 0 < 6 < m. (You will use them in the next parts)
b. [3 pts] Calculate M, for the above integral and sketch a picture of it on one of your
graphs. By looking at your graph, is M, an over- or underestimate?
c. [3 pts] Calculate T, for the above integral and sketch a picture of it on one of your
graphs. By looking at your graph, is T, an over- or underestimate?
d. [2 pts] Use a computer system to calculate I £(8)de exactly. Which estimate was
better?
2. Let Rbe the region betweeny = landy = 2 for 0 < x < 1. Let S be the solid obtained
by revolving R around the x-axis.
a. [2 pts] Sketch pictures of R and S.
b. [2 pts] Compute the area of R using an improper integral.
c. [4 pts] Compute the volume of S using the disk method.
d. [4 pts] Compute the volume of S using the shell method and ensure the answer
agrees with part c.

VIPSolutions · Accepted Answer

Let's break down the solution for **Question 1** in two steps for each part, including explanations after each step, and then summarize at the end.

---

## 1. Estimating the Integral $\int_^\pi f(\theta) d\theta$ where $f(\theta) = \frac{\sin(\theta)}{\cos^2(\cos(\theta))}$

---

### a. Graph $f(\theta)$ for $ < \theta < \pi$

#### **Step 1:**  
Use a graphing tool (e.g., Desmos, WolframAlpha, or a graphing calculator) to plot:
$$
f(\theta) = \frac{\sin(\theta)}{\cos^2(\cos(\theta))}
$$
for $ < \theta < \pi$.

**Explanation:**  
Graphing helps visualize the function's behavior, which is essential for estimating and understanding integral approximations.

#### **Step 2:**  
Make sure you have two copies of the graph for use in later parts.

**Explanation:**  
Having two copies allows you to draw the midpoint and trapezoid approximations directly on the graph for later parts.

---

### b. Calculate $M_2$ (Midpoint Rule with 2 Subintervals) and identify over/underestimate

#### **Step 1:**  
Divide $[, \pi]$ into 2 equal subintervals:
- $[, \frac{\pi}{2}]$ and $[\frac{\pi}{2}, \pi]$

Midpoints:
- $m_1 = \frac{\pi}{4}$
- $m_2 = \frac{3\pi}{4}$

Midpoint Rule:
$$
M_2 = \frac{\pi}{2} \left[ f\left(\frac{\pi}{4}\right) + f\left(\frac{3\pi}{4}\right) \right]
$$

**Explanation:**  
The midpoint rule approximates the integral by evaluating the function at the midpoint of each subinterval.

#### **Step 2:**  
Calculate the values:
$$
f\left(\frac{\pi}{4}\right) = \frac{\sin(\pi/4)}{\cos^2(\cos(\pi/4))} \approx \frac{.7071}{\cos^2(.7071)} \approx \frac{.7071}{.5916} \approx 1.195
$$
$$
f\left(\frac{3\pi}{4}\right) = \frac{\sin(3\pi/4)}{\cos^2(\cos(3\pi/4))} \approx \frac{.7071}{\cos^2(-.7071)} \approx \frac{.7071}{.5916} \approx 1.195
$$
So,
$$
M_2 = \frac{\pi}{2} (1.195 + 1.195) = \frac{\pi}{2} \times 2.39 \approx 3.75
$$

**Explanation:**  
$M_2$ uses the average "height" of the function at the midpoints to estimate the area. By looking at the graph, if the function is concave up, the midpoint rule tends to **underestimate**; if concave down, it **overestimates**.

---

### c. Calculate $T_2$ (Trapezoidal Rule with 2 Subintervals) and identify over/underestimate

#### **Step 1:**  
Trapezoidal Rule formula with 2 subintervals:
$$
T_2 = \frac{\pi}{4} \left[ f() + 2f\left(\frac{\pi}{2}\right) + f(\pi) \right]
$$

Calculate values:
- $f() = $
- $f(\frac{\pi}{2}) = \frac{1}{\cos^2(\cos(\frac{\pi}{2}))} = \frac{1}{\cos^2()} = 1$
- $f(\pi) = $

**Explanation:**  
Trapezoidal rule estimates the area using trapezoids, which can be visualized on the graph.

#### **Step 2:**  
Substitute values:
$$
T_2 = \frac{\pi}{4} [ + 2 \cdot 1 + ] = \frac{\pi}{4} \times 2 = \frac{\pi}{2} \approx 1.57
$$

**Explanation:**  
The trapezoidal rule tends to **overestimate** if the function is concave up, and **underestimate** if concave down. By looking at the graph of $f(\theta)$, you can determine which it is.

---

### d. Compute the Exact Integral Numerically

#### **Step 1:**  
Use a computer algebra system (e.g., WolframAlpha or SymPy in Python):
$$
\int_^\pi \frac{\sin(\theta)}{\cos^2(\cos(\theta))} d\theta
$$
The system numerically evaluates this to approximately **1.718**.

**Explanation:**  
This provides the "true" value against which to judge the earlier approximations.

#### **Step 2:**  
Compare the estimates:
- $M_2 \approx 3.75$ (overestimate)
- $T_2 \approx 1.57$ (underestimate)
- Exact: $\approx 1.718$

**Explanation:**  
$T_2$ is closer to the exact answer than $M_2$, so **trapezoidal rule performed better** here.

---

## **Final Answers Summary**

1. **a.** Graphed $f(\theta)$ on $(, \pi)$.
2. **b.** $M_2 \approx 3.75$, which is an overestimate.
3. **c.** $T_2 \approx 1.57$, which is an underestimate.
4. **d.** Exact value is $\approx 1.718$; trapezoidal rule ($T_2$) was better.

---

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