Question

Math

Posted 6 months ago

```
The graph $h$ is shown below.
Evaluate the definite integral of $\int_{-9}^{-3} h(x) d x$.
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Answer from Sia

Posted 6 months ago

Solution by Steps

step 1

To evaluate the definite integral $\int_{-9}^{-3} h(x) \, dx$, we need to find the area under the curve of $h(x)$ from $x = -9$ to $x = -3$

step 2

From the graph, we observe that $h(x)$ is a piecewise linear function. We need to break the integral into segments where $h(x)$ is linear

step 3

The first segment is from $x = -9$ to $x = -8$. The function $h(x)$ is linear and can be described by the line connecting the points $(-12, 0)$ and $(-8, 5)$

step 4

The second segment is from $x = -8$ to $x = -6$. The function $h(x)$ is linear and can be described by the line connecting the points $(-8, 5)$ and $(-6, 0)$

step 5

The third segment is from $x = -6$ to $x = -3$. The function $h(x)$ is linear and can be described by the line connecting the points $(-6, 0)$ and $(-3, -4)$

step 6

Calculate the area of each segment using the formula for the area of a trapezoid: $\text{Area} = \frac{1}{2} \times (\text{Base}_1 + \text{Base}_2) \times \text{Height}$

step 7

For the segment from $x = -9$ to $x = -8$: $\text{Base}_1 = h(-9) = 0$, $\text{Base}_2 = h(-8) = 5$, and $\text{Height} = 1$. So, $\text{Area}_1 = \frac{1}{2} \times (0 + 5) \times 1 = 2.5$

step 8

For the segment from $x = -8$ to $x = -6$: $\text{Base}_1 = h(-8) = 5$, $\text{Base}_2 = h(-6) = 0$, and $\text{Height} = 2$. So, $\text{Area}_2 = \frac{1}{2} \times (5 + 0) \times 2 = 5$

step 9

For the segment from $x = -6$ to $x = -3$: $\text{Base}_1 = h(-6) = 0$, $\text{Base}_2 = h(-3) = -4$, and $\text{Height} = 3$. So, $\text{Area}_3 = \frac{1}{2} \times (0 + -4) \times 3 = -6$

step 10

Sum the areas of all segments to find the total area under the curve from $x = -9$ to $x = -3$: $\text{Total Area} = 2.5 + 5 - 6 = 1.5$

Answer

The definite integral $\int_{-9}^{-3} h(x) \, dx$ is $1.5$.

Key Concept

Definite Integral of a Piecewise Linear Function

Explanation

To evaluate the definite integral of a piecewise linear function, break the integral into segments where the function is linear, calculate the area of each segment, and sum the areas.

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