Choose the one alternative that best completes the statement or answers the question.
Approximate the area under the curve and above the x-axis using n rectangles. Let the height of each rectangle be given
by the value of the function at the right side of the rectangle.
-$$f ( x ) = 2 x ^ { 2 } + x + 3 \text { from } x = - 2 \text { to } x = 1 ; n = 6$$
A) 7
B) 9
C) 11
D) 13

Question

Quizplus · Accepted Answer

To approximate the area under the curve $f(x) = 2x^2 + x + 3$ from $x = -2$ to $x = 1$ using $n = 6$ rectangles with the height of each rectangle given by the value of the function at the right side of the rectangle, we first calculate the width of each rectangle ($\Delta x$) and then sum up the areas of all rectangles.The total interval length is $1 - (-2) = 3$, so the width of each rectangle is $\Delta x = \frac{3}{6} = 0.5$.The x-values at the right side of each rectangle are: $-1.5, -1, -0.5, 0, 0.5, 1$.The area of each rectangle is $f(x_i) \cdot \Delta x$, where $x_i$ is the x-value at the right side of the i-th rectangle.Calculating the area for each rectangle and summing them up:1. $f(-1.5) = 2(-1.5)^2 + (-1.5) + 3 = 4.5 - 1.5 + 3 = 6$2. $f(-1) = 2(-1)^2 + (-1) + 3 = 2 - 1 + 3 = 4$3. $f(-0.5) = 2(-0.5)^2 + (-0.5) + 3 = 0.5 - 0.5 + 3 = 3$4. $f(0) = 2(0)^2 + (0) + 3 = 3$5. $f(0.5) = 2(0.5)^2 + (0.5) + 3 = 0.5 + 0.5 + 3 = 4$6. $f(1) = 2(1)^2 + (1) + 3 = 2 + 1 + 3 = 6$The total area is approximately the sum of these areas times the width ($\Delta x = 0.5$):Total area $\approx (6 + 4 + 3 + 3 + 4 + 6) \cdot 0.5 = 13$.Therefore, the correct answer is D) 13.

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