Question

Math

Posted 2 months ago

```
Approximate the area between $h(x)$ and the $x$-axis from $x=-1$ to $x=1$ using a right Riemann sum with 4 equal subdivisions.
$R(4)=$
$\square$ units $^{2}$.
```

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Answer from Sia

Posted 2 months ago

Solution by Steps

step 1

To approximate the area between $h(x)$ and the $x$-axis from $x = -1$ to $x = 1$ using a right Riemann sum with 4 equal subdivisions, we first determine the width of each subdivision. The interval $[-1, 1]$ is divided into 4 equal parts, so each subdivision has a width of $\Delta x = \frac{1 - (-1)}{4} = \frac{2}{4} = 0.5$

step 2

Next, we identify the right endpoints of each subdivision. These are $x = -0.5, 0, 0.5, 1$

step 3

We then evaluate the function $h(x)$ at these right endpoints: $h(-0.5), h(0), h(0.5), h(1)$

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:
$R(4) = \Delta x \left( h(-0.5) + h(0) + h(0.5) + h(1) \right)$
$R(4) = 0.5 \left( h(-0.5) + h(0) + h(0.5) + h(1) \right)$

step 5

Substitute the values of $h(x)$ at the right endpoints into the formula to get the final approximation

Answer

$R(4) = 0.5 \left( h(-0.5) + h(0) + h(0.5) + h(1) \right)$ units$^2$

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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