QuestionJuly 14, 2025

Approximate the area under the graph of f(x) and above the x-axis with rectangles, using the following methods with n=4 f(x)=e^x+4 from x=-2 to x=2 (a) Use left endpoints. (b) Use right endpoints. (c) Average the answers in parts (a) and (b) (d) Use midpoints.

Approximate the area under the graph of f(x) and above the x-axis with rectangles, using the following methods with n=4 f(x)=e^x+4 from x=-2 to x=2 (a) Use left endpoints. (b) Use right endpoints. (c) Average the answers in parts (a) and (b) (d) Use midpoints.
Approximate the area under the graph of f(x) and above the x-axis with rectangles, using the following methods with n=4 f(x)=e^x+4 from x=-2 to x=2 (a) Use left endpoints. (b) Use right endpoints. (c) Average the answers in parts (a) and (b) (d) Use midpoints.

Solution
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Answer

(a) A_L = 19.548 ### (b) A_R = 46.778 ### (c) A_{avg} = 33.163 ### (d) A_M = 30.663 Explanation 1. Calculate the width of each rectangle The interval is from x = -2 to x = 2, so the total width is 4. With n = 4, the width of each rectangle is \Delta x = \frac{4}{4} = 1. 2. Left endpoint approximation Evaluate f(x) at left endpoints: x = -2, -1, 0, 1. - f(-2) = e^{-2} + 4 - f(-1) = e^{-1} + 4 - f(0) = e^{0} + 4 = 5 - f(1) = e^{1} + 4 Sum the areas: A_L = \Delta x [f(-2) + f(-1) + f(0) + f(1)]. 3. Right endpoint approximation Evaluate f(x) at right endpoints: x = -1, 0, 1, 2. - f(-1) = e^{-1} + 4 - f(0) = e^{0} + 4 = 5 - f(1) = e^{1} + 4 - f(2) = e^{2} + 4 Sum the areas: A_R = \Delta x [f(-1) + f(0) + f(1) + f(2)]. 4. Average of left and right approximations Compute average: A_{avg} = \frac{A_L + A_R}{2}. 5. Midpoint approximation Evaluate f(x) at midpoints: x = -1.5, -0.5, 0.5, 1.5. - f(-1.5) = e^{-1.5} + 4 - f(-0.5) = e^{-0.5} + 4 - f(0.5) = e^{0.5} + 4 - f(1.5) = e^{1.5} + 4 Sum the areas: A_M = \Delta x [f(-1.5) + f(-0.5) + f(0.5) + f(1.5)].

Explanation

1. Calculate the width of each rectangle<br /> The interval is from $x = -2$ to $x = 2$, so the total width is $4$. With $n = 4$, the width of each rectangle is $\Delta x = \frac{4}{4} = 1$.<br /><br />2. Left endpoint approximation<br /> Evaluate $f(x)$ at left endpoints: $x = -2, -1, 0, 1$. <br />- $f(-2) = e^{-2} + 4$<br />- $f(-1) = e^{-1} + 4$<br />- $f(0) = e^{0} + 4 = 5$<br />- $f(1) = e^{1} + 4$<br /><br /> Sum the areas: $A_L = \Delta x [f(-2) + f(-1) + f(0) + f(1)]$.<br /><br />3. Right endpoint approximation<br /> Evaluate $f(x)$ at right endpoints: $x = -1, 0, 1, 2$.<br />- $f(-1) = e^{-1} + 4$<br />- $f(0) = e^{0} + 4 = 5$<br />- $f(1) = e^{1} + 4$<br />- $f(2) = e^{2} + 4$<br /><br /> Sum the areas: $A_R = \Delta x [f(-1) + f(0) + f(1) + f(2)]$.<br /><br />4. Average of left and right approximations<br /> Compute average: $A_{avg} = \frac{A_L + A_R}{2}$.<br /><br />5. Midpoint approximation<br /> Evaluate $f(x)$ at midpoints: $x = -1.5, -0.5, 0.5, 1.5$.<br />- $f(-1.5) = e^{-1.5} + 4$<br />- $f(-0.5) = e^{-0.5} + 4$<br />- $f(0.5) = e^{0.5} + 4$<br />- $f(1.5) = e^{1.5} + 4$<br /><br /> Sum the areas: $A_M = \Delta x [f(-1.5) + f(-0.5) + f(0.5) + f(1.5)]$.
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