Use left endpoints and 10 rectangles to find the approximation of the area of the region between the graph of the function $4 x ^ { 2 } - x - 1 \text { and the } x \text {-axis over the interval } ( 4,14 ) \text {. Round }$ your answer to the nearest integer.
A) 2925
B) 3325
C) 3000
D) 3250
E) 3125

Question

Use left endpoints and 10 rectangles to find the approximation of the area of the region between the graph of the function $4 x ^ { 2 } - x - 1 \text { and the } x \text {-axis over the interval } ( 4,14 ) \text {. Round }$ your answer to the nearest integer. 
A) 2925
B) 3325
C) 3000
D) 3250
E) 3125

Quizplus · Accepted Answer

The interval from 4 to 14 has a length of 10. Dividing this interval into 10 rectangles means each rectangle has a width ($\Delta x$) of 1. Using the left endpoint for each rectangle, the x-values are 4, 5, ..., 13. The height of each rectangle is given by the function $f(x) = 4x^2 - x - 1$. The area of each rectangle is $f(x_i) \times \Delta x$, and the total area is the sum of the areas of these rectangles. Calculating this sum:$$\begin{align*}\text{Area} & = \sum_{i=4}^{13} (4x_i^2 - x_i - 1)(1) \& = (4(4)^2 - 4 - 1) + (4(5)^2 - 5 - 1) + \cdots + (4(13)^2 - 13 - 1) \& = 59 + 96 + 139 + 188 + 243 + 304 + 371 + 444 + 523 + 608 \& = 3125.\end{align*}$$Therefore, the approximation of the area using left endpoints and 10 rectangles is 3125, rounded to the nearest integer.

Use Left Endpoints and 10 Rectangles to Find the Approximation 4x2−x−1 and the x-axis over the interval (4,14). Round 4 x ^ { 2 } - x - 1 \text { and the } x \text {-axis over the interval } ( 4,14 ) \text {. Round }4x2−x−1 and the x-axis over the interval (4,14). Round

Use Left Endpoints and 10 Rectangles to Find the Approximation $4 x ^ { 2 } - x - 1 \text { and the } x \text {-axis over the interval } ( 4,14 ) \text {. Round }$