To estimate the area under the curve $f(x) = 5\cos(x)$ from $x = 0$ to $x = \pi/2$ using four approximating rectangles with right endpoints, follow these steps:

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Step 1: Determine the width of each rectangle ($\Delta x$).

The interval $[0, \pi/2]$ is divided into $n = 4$ equal subintervals:

$\Delta x = \frac{\pi/2 - 0}{4} = \frac{\pi}{8}.$

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Step 2: Identify the right endpoints.

The right endpoints for the subintervals are: $x_1 = \frac{\pi}{8}, \, x_2 = \frac{2\pi}{8}, \, x_3 = \frac{3\pi}{8}, \, x_4 = \frac{4\pi}{8} = \frac{\pi}{2}.$

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Step 3: Evaluate $f(x)$ at the right endpoints.

The function is $f(x) = 5\cos(x)$. Calculate $f(x)$ at each right endpoint:

1. $f(x_1) = 5\cos\left(\frac{\pi}{8}\right)$,
2. $f(x_2) = 5\cos\left(\frac{2\pi}{8}\right)$,
3. $f(x_3) = 5\cos\left(\frac{3\pi}{8}\right)$,
4. $f(x_4) = 5\cos\left(\frac{4\pi}{8}\right) = 5\cos\left(\frac{\pi}{2}\right) = 0.$

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Step 4: Apply the formula for $R_4$.

The formula for the Riemann sum with right endpoints is:

$R_4 = \Delta x \cdot \left[ f(x_1) + f(x_2) + f(x_3) + f(x_4) \right].$

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Step 5: Perform the calculations.

First, approximate the values of $\cos(x)$:

• $\cos\left(\frac{\pi}{8}\right) \approx 0.9239$,
• $\cos\left(\frac{2\pi}{8}\right) = \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \approx 0.7071$,
• $\cos\left(\frac{3\pi}{8}\right) \approx 0.3827$,
• $\cos\left(\frac{\pi}{2}\right) = 0$.

Now substitute into $f(x) = 5\cos(x)$:

• $f(x_1) = 5 \cdot 0.9239 = 4.6195$,
• $f(x_2) = 5 \cdot 0.7071 = 3.5355$,
• $f(x_3) = 5 \cdot 0.3827 = 1.9135$,
• $f(x_4) = 5 \cdot 0 = 0.$.

Sum these values:

$f(x_1) + f(x_2) + f(x_3) + f(x_4) = 4.6195 + 3.5355 + 1.9135 + 0 = 10.0685.$

Finally, multiply by $\Delta x = \frac{\pi}{8} \approx 0.3927$:

$R_4 = 0.3927 \cdot 10.0685 \approx 3.9515.$

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Final Answer:

$R_4 \approx 3.9515$

Let me know if you'd like further clarification or details!

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Related Questions:

1. What is the formula for a Riemann sum with left endpoints?
2. How does increasing the number of subintervals affect the accuracy of the approximation?
3. How is the trapezoidal rule different from Riemann sums?
4. What is the exact integral value of $\int_0^{\pi/2} 5\cos(x) \, dx$?
5. Can midpoint Riemann sums provide a more accurate result than right or left endpoint sums?

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Tip:

When approximating integrals, visualize the graph of the function to understand how the rectangles relate to the area under the curve!