At least one of the answers above is NOT correct.
(1 point)
Suppose .
(a) The rectangles in the graph on the left illustrate a left endpoint Riemann sum for on the interval . The value of this left endpoint Riemann sum is , and it is an underestimate of the area of the region enclosed by , the -axis, and the vertical lines and .
(b) The rectangles in the graph on the right illustrate a right endpoint Riemann sum for on the interval . The value of this right endpoint Riemann sum is , and it is an overestimate of the area of the region enclosed by , the -axis, and the vertical lines and .
Preview Mv Answers
Submit Answers

Ask by Pearson Mcfarlane. in the United States

Mar 27,2025

Question

At least one of the answers above is NOT correct.
(1 point)
Suppose .
(a) The rectangles in the graph on the left illustrate a left endpoint Riemann sum for on the interval . The value of this left endpoint Riemann sum is , and it is an underestimate of the area of the region enclosed by , the -axis, and the vertical lines and .
(b) The rectangles in the graph on the right illustrate a right endpoint Riemann sum for on the interval . The value of this right endpoint Riemann sum is , and it is an overestimate of the area of the region enclosed by , the -axis, and the vertical lines and .
Preview Mv Answers
Submit Answers

Ask by Pearson Mcfarlane. in the United States

Mar 27,2025

UpStudy AI · Accepted Answer

We begin with the function

$$
f(x)=\frac{x^2}{10},
$$

defined on the interval $$[2,4]$$. Because the problem shows Riemann sums (and the graphs usually indicate the number of rectangles), we will assume the standard setup using two rectangles. (This is the most common case when no further information is provided.)

### Step 1. Determine the Width of Each Subinterval

Since the interval is $$[2,4]$$ and we use two rectangles, the width of each subinterval is

$$
\Delta x = \frac{4-2}{2}=1.
$$

### Step 2. Compute the Left Endpoint Riemann Sum

For a left endpoint Riemann sum with two rectangles, the sample points are the left endpoints of the subintervals. The subintervals and their left endpoints are:

- For the first rectangle on $$[2,3]$$: the left endpoint is $$x=2$$.
- For the second rectangle on $$[3,4]$$: the left endpoint is $$x=3$$.

Now compute the function values at these endpoints:

$$
f(2)=\frac{2^2}{10}=\frac{4}{10},
$$
$$
f(3)=\frac{3^2}{10}=\frac{9}{10}.
$$

Thus, the left endpoint Riemann sum is

$$
L = f(2)\Delta x + f(3)\Delta x = \frac{4}{10}\cdot 1 + \frac{9}{10}\cdot 1 = \frac{4+9}{10}=\frac{13}{10}.
$$

Because $$f(x)=\frac{x^2}{10}$$ is an increasing function, the left endpoint sum gives a lower (under) estimate of the true area.

### Step 3. Compute the Right Endpoint Riemann Sum

For a right endpoint Riemann sum with two rectangles, the sample points are the right endpoints of the subintervals. The subintervals and their right endpoints are:

- For the first rectangle on $$[2,3]$$: the right endpoint is $$x=3$$.
- For the second rectangle on $$[3,4]$$: the right endpoint is $$x=4$$.

Now compute the function values at these endpoints:

$$
f(3)=\frac{3^2}{10}=\frac{9}{10},
$$
$$
f(4)=\frac{4^2}{10}=\frac{16}{10}.
$$

Thus, the right endpoint Riemann sum is

$$
R = f(3)\Delta x + f(4)\Delta x = \frac{9}{10}\cdot 1 + \frac{16}{10}\cdot 1 = \frac{9+16}{10}=\frac{25}{10}=\frac{5}{2}.
$$

Since the function is increasing, the right endpoint gives an overestimate of the true area.

### Final Answers

(a) The left endpoint Riemann sum is $$\frac{13}{10}$$ and it is an **underestimate** of the area.

(b) The right endpoint Riemann sum is $$\frac{5}{2}$$ and it is an **overestimate** of the area.
(a) The left endpoint Riemann sum is $$ \frac{13}{10} $$ and it is an underestimate of the area. (b) The right endpoint Riemann sum is $$ \frac{5}{2} $$ and it is an overestimate of the area.

Solución de inteligencia artificial de Upstudy

Responder

Solución

Bonus Knowledge

preguntas relacionadas

Latest Calculus Questions