Let's go through each part of the problem step-by-step:

Problem Summary

You are given the function: $f(x) = 4x + 1$ and asked to estimate the area between the graph of $f(x)$ and the x-axis from $x = 0$ to $x = 3$ using six rectangles.

1. Part (a) asks for the area estimate using right endpoints.
2. Part (b) asks for the area estimate using left endpoints.

Step-by-Step Solution

1. Determine the Width of Each Rectangle:

   • The interval from $x = 0$ to $x = 3$ has length 3.
   • Since we are using six rectangles, each rectangle will have a width of: $\Delta x = \frac{3 - 0}{6} = 0.5$

2. Right Endpoint Approximation (Part (a)):

   • With a width of $\Delta x = 0.5$, the x-values for the right endpoints of each rectangle will be $x = 0.5, 1.0, 1.5, 2.0, 2.5,$ and $3.0$.
   • Calculate $f(x)$ at each of these right endpoints:
     • $f(0.5) = 4(0.5) + 1 = 2 + 1 = 3$
     • $f(1.0) = 4(1.0) + 1 = 4 + 1 = 5$
     • $f(1.5) = 4(1.5) + 1 = 6 + 1 = 7$
     • $f(2.0) = 4(2.0) + 1 = 8 + 1 = 9$
     • $f(2.5) = 4(2.5) + 1 = 10 + 1 = 11$
     • $f(3.0) = 4(3.0) + 1 = 12 + 1 = 13$
   • The area estimate using right endpoints is the sum of the areas of the six rectangles: $\text{Area}_{\text{right}} = \Delta x \times (f(0.5) + f(1.0) + f(1.5) + f(2.0) + f(2.5) + f(3.0))$ $= 0.5 \times (3 + 5 + 7 + 9 + 11 + 13) = 0.5 \times 48 = 24$

3. Left Endpoint Approximation (Part (b)):

   • The x-values for the left endpoints of each rectangle are $x = 0, 0.5, 1.0, 1.5, 2.0,$ and $2.5$.
   • Calculate $f(x)$ at each of these left endpoints:
     • $f(0) = 4(0) + 1 = 1$
     • $f(0.5) = 4(0.5) + 1 = 3$
     • $f(1.0) = 4(1.0) + 1 = 5$
     • $f(1.5) = 4(1.5) + 1 = 7$
     • $f(2.0) = 4(2.0) + 1 = 9$
     • $f(2.5) = 4(2.5) + 1 = 11$
   • The area estimate using left endpoints is: $\text{Area}_{\text{left}} = \Delta x \times (f(0) + f(0.5) + f(1.0) + f(1.5) + f(2.0) + f(2.5))$ $= 0.5 \times (1 + 3 + 5 + 7 + 9 + 11) = 0.5 \times 36 = 18$

Final Answers

• (a) Using right endpoints: $\text{Area}_{\text{right}} = 24.00$
• (b) Using left endpoints: $\text{Area}_{\text{left}} = 18.00$

Would you like additional details or have any questions?

Related Questions

1. How does changing the number of rectangles affect the area approximation?
2. How would you estimate the area using midpoint endpoints?
3. How can you calculate the exact area under $f(x) = 4x + 1$ from $x = 0$ to $x = 3$ using integration?
4. What is the significance of using right versus left endpoints in approximations?
5. How would you use a trapezoidal approximation for this problem?

Tip

When approximating areas under a curve, more rectangles generally lead to a more accurate estimate, especially as they get narrower.