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Question
Math
Posted 2 months ago

Approximate the area between h(x)h(x) and the xx-axis from x=1x=-1 to x=1x=1 using a right Riemann sum with 4 equal subdivisions.
R(4)=
R(4)=

\square units 2^{2}.
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Answer from Sia
Posted 2 months ago
Solution by Steps
step 1
To approximate the area between h(x)h(x) and the xx-axis from x=1x = -1 to x=1x = 1 using a right Riemann sum with 4 equal subdivisions, we first determine the width of each subdivision. The interval [1,1][-1, 1] is divided into 4 equal parts, so each subdivision has a width of Δx=1(1)4=24=0.5\Delta x = \frac{1 - (-1)}{4} = \frac{2}{4} = 0.5
step 2
Next, we identify the right endpoints of each subdivision. These are x=0.5,0,0.5,1x = -0.5, 0, 0.5, 1
step 3
We then evaluate the function h(x)h(x) at these right endpoints: h(0.5),h(0),h(0.5),h(1)h(-0.5), h(0), h(0.5), h(1)
step 4
The right Riemann sum is given by the sum of the function values at these points, each multiplied by the width of the subdivisions: R(4)=Δx(h(0.5)+h(0)+h(0.5)+h(1)) R(4) = \Delta x \left( h(-0.5) + h(0) + h(0.5) + h(1) \right) R(4)=0.5(h(0.5)+h(0)+h(0.5)+h(1)) R(4) = 0.5 \left( h(-0.5) + h(0) + h(0.5) + h(1) \right)
step 5
Substitute the values of h(x)h(x) at the right endpoints into the formula to get the final approximation
Answer
R(4)=0.5(h(0.5)+h(0)+h(0.5)+h(1))R(4) = 0.5 \left( h(-0.5) + h(0) + h(0.5) + h(1) \right) units2^2
Key Concept
Right Riemann Sum
Explanation
The right Riemann sum approximates the area under a curve by summing the areas of rectangles whose heights are determined by the function values at the right endpoints of each subdivision.

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