Estimate the area under the graph of

Question

Estimate the area under the graph of 
f(x) = 3 cos(x)
 from 
x = 0
 to 
x = 
𝜋
2
 using four approximating rectangles and right endpoints, sketch the graph and the rectangles, and determine if your estimate is an underestimate or an overestimate.

GPT-4o mini · Answer

To estimate the area under the graph of \( f(x) = 3 \cos(x) \) from \( x = 0 \) to \( x = \frac{\pi}{2} \) using four approximating rectangles with right endpoints, follow these steps:

Step 1: Calculate the width of each rectangle

The width \( \Delta x \) of each rectangle is given by: \[ \Delta x = \frac{b - a}{n} \] where \( a = 0 \), \( b = \frac{\pi}{2} \), and \( n = 4 \).

Calculating this: \[ \Delta x = \frac{\frac{\pi}{2} - 0}{4} = \frac{\pi}{8} \]

Step 2: Determine the right endpoints

The right endpoints \( x_i \) for \( i = 1, 2, 3, 4 \) are calculated as:

For \( i = 1 \): \( x_1 = 0 + \Delta x = 0 + \frac{\pi}{8} = \frac{\pi}{8} \)
For \( i = 2 \): \( x_2 = \frac{\pi}{8} + \Delta x = \frac{\pi}{8} + \frac{\pi}{8} = \frac{2\pi}{8} = \frac{\pi}{4} \)
For \( i = 3 \): \( x_3 = \frac{2\pi}{8} + \Delta x = \frac{3\pi}{8} \)
For \( i = 4 \): \( x_4 = \frac{3\pi}{8} + \Delta x = \frac{4\pi}{8} = \frac{\pi}{2} \)

The right endpoints are:

\( x_1 = \frac{\pi}{8} \)
\( x_2 = \frac{\pi}{4} \)
\( x_3 = \frac{3\pi}{8} \)
\( x_4 = \frac{\pi}{2} \)

Step 3: Evaluate the function at each right endpoint

Calculating \( f(x_i) \):

\( f(x_1) = 3 \cos\left(\frac{\pi}{8}\right) \)
\( f(x_2) = 3 \cos\left(\frac{\pi}{4}\right) = 3 \cdot \frac{\sqrt{2}}{2} = \frac{3\sqrt{2}}{2} \)
\( f(x_3) = 3 \cos\left(\frac{3\pi}{8}\right) \)
\( f(x_4) = 3 \cos\left(\frac{\pi}{2}\right) = 3 \cdot 0 = 0 \)

Step 4: Estimate the area using right endpoints

Now, we can estimate the area \( A \) using the formula: \[ A \approx \sum_{i=1}^{n} f(x_i) \Delta x \]

Calculating the individual areas: \[ A \approx (3 \cos\left(\frac{\pi}{8}\right) + \frac{3\sqrt{2}}{2} + 3 \cos\left(\frac{3\pi}{8}\right) + 0) \cdot \frac{\pi}{8} \]

Using cosine values:

\( \cos\left(\frac{\pi}{8}\right) \approx 0.92388 \)
\( \cos\left(\frac{3\pi}{8}\right) \approx 0.38268 \)

Plugging in the values: \[ f(x_1) \approx 3 \cdot 0.92388 \approx 2.77164 \] \[ f(x_3) \approx 3 \cdot 0.38268 \approx 1.14804 \]

Now substituting these values: \[ A \approx (2.77164 + 2.12132 + 1.14804 + 0) \cdot \frac{\pi}{8} \] Calculating the sum: \[ A \approx (6.04100) \cdot \frac{\pi}{8} \approx \frac{6.04100 \pi}{8} \approx 0.75513 \pi \approx 2.369 \text{ (approximately)} \]

Conclusion: Underestimate or Overestimate?

Since \( f(x) = 3 \cos(x) \) is decreasing on the interval \( [0, \frac{\pi}{2}] \), using right endpoints will yield an overestimate. Thus, our estimate of the area using right endpoints with four rectangles is an overestimate.

Sketch

To sketch the graph:

Draw the curve for \( f(x) = 3 \cos(x) \) from \( x = 0 \) to \( x = \frac{\pi}{2} \).
Draw the rectangles starting from \( x_1 \) to \( x_2 \), \( x_2 \) to \( x_3 \), \( x_3 \) to \( x_4 \).
The height of each rectangle is given by \( f(x_i) \) where \( x_i \) is the right endpoint.

Illustratively, the graph should show a smooth decrease from \( f(0) = 3 \) to \( f\left(\frac{\pi}{2}\right) = 0\), as well as the rectangles extending to the values found from the evaluation at the right endpoints.