Question

1. (15 points)

a. estimate the area under the graph of $f(x) = x^2 + x$ from $x = 0$ to $x = 3$ using 3 approximating rectangles and left endpoints.

Explanation:

Step1: Determine the width of each rectangle

The interval is from \( x = 0 \) to \( x = 3 \), and we are using \( n = 3 \) rectangles. The width \( \Delta x \) of each rectangle is calculated by the formula \( \Delta x=\frac{b - a}{n} \), where \( a = 0 \), \( b = 3 \), and \( n = 3 \).
\[
\Delta x=\frac{3 - 0}{3}=1
\]

Step2: Identify the left endpoints

The left endpoints for the three rectangles (since \( \Delta x = 1 \)) will be at \( x_0 = 0 \), \( x_1 = 1 \), and \( x_2 = 2 \).

Step3: Calculate the height of each rectangle at the left endpoints

We use the function \( f(x)=x^{2}+x \) to find the height of each rectangle at the left endpoints.

• For \( x_0 = 0 \): \( f(0)=0^{2}+0 = 0 \)
• For \( x_1 = 1 \): \( f(1)=1^{2}+1 = 2 \)
• For \( x_2 = 2 \): \( f(2)=2^{2}+2 = 6 \)

Step4: Calculate the area of each rectangle and sum them up

The area of each rectangle is \( \text{height}\times\Delta x \). We sum up the areas of the three rectangles.
\[
\text{Area}\approx f(x_0)\Delta x + f(x_1)\Delta x + f(x_2)\Delta x
\]
Substitute the values we found:
\[
\text{Area}\approx(0\times1)+(2\times1)+(6\times1)=0 + 2+6 = 8
\]

Answer:

The estimated area under the graph is \( 8 \).