To approximate the integral $\int_{4}^{19} f(x) \, dx$ using the right Riemann sum with the given 5 subintervals, we'll follow these steps:

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Step-by-Step Solution:

1. Subintervals: From the table, the $x$-values divide the interval $[4, 19]$ into the following 5 subintervals:

   • $[4, 7]$
   • $[7, 11]$
   • $[11, 13]$
   • $[13, 16]$
   • $[16, 19]$

2. Widths ($\Delta x$): The width of each subinterval is the difference between consecutive $x$-values:

   • $\Delta x_1 = 7 - 4 = 3$
   • $\Delta x_2 = 11 - 7 = 4$
   • $\Delta x_3 = 13 - 11 = 2$
   • $\Delta x_4 = 16 - 13 = 3$
   • $\Delta x_5 = 19 - 16 = 3$

3. Function Values ($f(x)$): For the right Riemann sum, we use the function values at the right endpoints of each subinterval:

   • For $[4, 7]$, the right endpoint is $x = 7$, so $f(7) = 18$
   • For $[7, 11]$, the right endpoint is $x = 11$, so $f(11) = 24$
   • For $[11, 13]$, the right endpoint is $x = 13$, so $f(13) = 29$
   • For $[13, 16]$, the right endpoint is $x = 16$, so $f(16) = 40$
   • For $[16, 19]$, the right endpoint is $x = 19$, so $f(19) = 28$

4. Right Riemann Sum Formula: The approximation is given by: $\text{Right Riemann Sum} = \sum_{i=1}^5 f(x_i^*) \cdot \Delta x_i$ Where $x_i^*$ is the right endpoint of the $i$-th subinterval.

5. Substitute Values: $\text{Right Riemann Sum} = (18 \cdot 3) + (24 \cdot 4) + (29 \cdot 2) + (40 \cdot 3) + (28 \cdot 3)$

6. Simplify Each Term:

   • $18 \cdot 3 = 54$
   • $24 \cdot 4 = 96$
   • $29 \cdot 2 = 58$
   • $40 \cdot 3 = 120$
   • $28 \cdot 3 = 84$

7. Sum Them Up: $\text{Right Riemann Sum} = 54 + 96 + 58 + 120 + 84 = 412$

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Final Answer:

The right Riemann sum approximation for $\int_{4}^{19} f(x) \, dx$ is: $\boxed{412}$

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Would you like a detailed explanation of any specific step? 😊

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5 Related Questions:

1. What is the difference between a right Riemann sum and a left Riemann sum?
2. How does the number of subintervals affect the accuracy of a Riemann sum?
3. How would the midpoint Riemann sum approximation differ for this example?
4. Can you calculate the average value of the function $f(x)$ over $[4, 19]$?
5. How is the trapezoidal rule different from the Riemann sum in approximating integrals?

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Tip:

When solving Riemann sums, always carefully identify the endpoints and widths of subintervals, as errors in these values can easily propagate through calculations!