(1 point) Estimate the area under the graph of from to using 4 approximating rectangles and left endpoints.
Approximation

Ask by Robbins Schofield. in the United States

Mar 27,2025

Question

(1 point) Estimate the area under the graph of $$ f(x)=x^{2}+3 x $$ from $$ x=3 $$ to $$ x=11 $$ using 4 approximating rectangles and left endpoints. Approximation $$ = $$

UpStudy AI · Accepted Answer

Let the number of rectangles be $$ n = 4 $$. The interval is from $$ x = 3 $$ to $$ x = 11 $$, so the width of each rectangle is

$$
\Delta x = \frac{11 - 3}{4} = 2.
$$

Since we are using left endpoints, the $$ x $$-values used will be:
$$
x_0 = 3, \quad x_1 = 3 + 2 = 5, \quad x_2 = 5 + 2 = 7, \quad x_3 = 7 + 2 = 9.
$$

Evaluate the function $$ f(x) = x^2 + 3x $$ at these points:

$$
f(3) = 3^2 + 3(3) = 9 + 9 = 18,
$$
$$
f(5) = 5^2 + 3(5) = 25 + 15 = 40,
$$
$$
f(7) = 7^2 + 3(7) = 49 + 21 = 70,
$$
$$
f(9) = 9^2 + 3(9) = 81 + 27 = 108.
$$

The area approximation is then

$$
\text{Area} \approx \Delta x \left[ f(3) + f(5) + f(7) + f(9) \right] = 2 \left( 18 + 40 + 70 + 108 \right).
$$

Calculating the sum inside the parentheses:

$$
18 + 40 + 70 + 108 = 236.
$$

Thus, the approximate area is

$$
2 \times 236 = 472.
$$

The final approximation is

$$
\boxed{472}.
$$
The approximate area under the graph is 472.

(1 point) Estimate the area under the graph of from to using 4 approximating rectangles and left endpoints.
Approximation

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(1 point) Estimate the area under the graph of from to using 4 approximating rectangles and left endpoints. Approximation