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Approximate the area between the
x-axis and
g(x)=2^(x) from
x=-2 to
x=2 using a right Riemann sum with 4 equal subdivisions.
The approximate area is
◻ units
^(2).
Here's a sketch to help you visualize the area:

Approximate the area between the $x$ -axis and $g(x)=2^{x}$ from $x=-2$ to $x=2$ using a right Riemann sum with $4$ equal subdivisions. $\newline$ The approximate area is $\square$ units $^{2}$ . $\newline$ Here's a sketch to help you visualize the area:

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Area of quadrilaterals and triangles: word problems

Full solution

Q. Approximate the area between the

x

-axis and

g(x)=2^{x}

from

x=-2

to

x=2

using a right Riemann sum with

4

equal subdivisions.

\newline

The approximate area is

\square

units

^{2}

.

\newline

Here's a sketch to help you visualize the area:

Calculate Subdivision Width: First, we need to find the width of each subdivision. The interval from $x=-2$ to $x=2$ is $4$ units wide. Since we're using $4$ subdivisions, each one will be $1$ unit wide. $\newline$ Width of each subdivision = $(2 - (-2)) / 4 = 4 / 4 = 1$
Calculate Right Endpoints: Next, we calculate the right endpoints for each subdivision. These are $x=-1$ , $x=0$ , $x=1$ , and $x=2$ .
Evaluate Function Values: Now we evaluate the function $g(x)$ at each of these right endpoints. $g(-1) = 2^{-1} = 0.5$ $g(0) = 2^{0} = 1$ $g(1) = 2^{1} = 2$ $g(2) = 2^{2} = 4$
Calculate Rectangle Areas: We multiply the function values by the width of each subdivision to approximate the area of each rectangle. $\newline$ Area of rectangle $1$ = $g(-1) \times \text{width} = 0.5 \times 1 = 0.5$ $\newline$ Area of rectangle $2$ = $g(0) \times \text{width} = 1 \times 1 = 1$ $\newline$ Area of rectangle $3$ = $g(1) \times \text{width} = 2 \times 1 = 2$ $\newline$ Area of rectangle $4$ = $g(2) \times \text{width} = 4 \times 1 = 4$
Calculate Total Approximate Area: Finally, we add up the areas of all rectangles to get the total approximate area under the curve. $\newline$ Total approximate area = Area of rectangle $1$ + Area of rectangle $2$ + Area of rectangle $3$ + Area of rectangle $4$ $\newline$ Total approximate area = $0.5 + 1 + 2 + 4 = 7.5$

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