Math  /  Geometry

QuestionPart 1 of 3
The graph below shows a rectangular sum of n=8n=8 rectangles to approximate the area under the line from x=0x=0 to x=2x=2.
Is this a right-hand or left-hand sum? right-hand sum σ\checkmark^{\vee} \sigma^{\infty} \square \qquad What is the equation of the line? y=2xy=2 x \quad Part 3 of 3
What is the value of the sum?

Studdy Solution

STEP 1

1. The graph represents a right-hand sum approximation.
2. The line equation is y=2x y = 2x .
3. The interval for the approximation is from x=0 x = 0 to x=2 x = 2 .
4. The number of rectangles n=8 n = 8 .

STEP 2

1. Determine the width of each rectangle.
2. Identify the right-hand endpoints for each rectangle.
3. Calculate the height of each rectangle using the line equation.
4. Compute the area of each rectangle.
5. Sum the areas of all rectangles to approximate the total area under the curve.

STEP 3

Determine the width of each rectangle:
The total interval length is 20=2 2 - 0 = 2 . With n=8 n = 8 rectangles, the width of each rectangle is:
Δx=28=0.25 \Delta x = \frac{2}{8} = 0.25

STEP 4

Identify the right-hand endpoints for each rectangle:
The right-hand endpoints are at x=0.25,0.5,0.75,,2.0 x = 0.25, 0.5, 0.75, \ldots, 2.0 .

STEP 5

Calculate the height of each rectangle using the line equation y=2x y = 2x :
For each right-hand endpoint xi x_i , the height is yi=2xi y_i = 2x_i .

STEP 6

Compute the area of each rectangle:
The area of each rectangle is Areai=yi×Δx=2xi×0.25 \text{Area}_i = y_i \times \Delta x = 2x_i \times 0.25 .

STEP 7

Sum the areas of all rectangles:
Total Area=i=18Areai=i=18(2xi×0.25)\text{Total Area} = \sum_{i=1}^{8} \text{Area}_i = \sum_{i=1}^{8} (2x_i \times 0.25)
Calculate each term:
=0.25×(2×0.25+2×0.5+2×0.75++2×2.0)= 0.25 \times (2 \times 0.25 + 2 \times 0.5 + 2 \times 0.75 + \ldots + 2 \times 2.0)
=0.25×(0.5+1.0+1.5+2.0+2.5+3.0+3.5+4.0)= 0.25 \times (0.5 + 1.0 + 1.5 + 2.0 + 2.5 + 3.0 + 3.5 + 4.0)
=0.25×18.0=4.5= 0.25 \times 18.0 = 4.5
The value of the sum is:
4.5 \boxed{4.5}

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