Question

Math

Posted 3 months ago

```
The graph of function $g$ is shown below. Let $h(x)=\int_{-5}^{x} g(t) d t$.
Evaluate $h(0)$.
$h(0)=$
```

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Answer from Sia

Posted 3 months ago

Solution by Steps

step 1

To evaluate $h(0)$, we need to compute the definite integral of $g(t)$ from $-5$ to $0$. This is given by $h(0) = \int_{-5}^{0} g(t) \, dt$

step 2

From the graph, we can see that $g(t)$ forms a triangle from $(-5,0)$ to $(-3,1)$ and then another triangle from $(-3,1)$ to $(0,0)$. We need to find the area of these triangles

step 3

The first triangle has a base of $2$ units (from $-5$ to $-3$) and a height of $1$ unit. The area of this triangle is $\frac{1}{2} \times 2 \times 1 = 1$

step 4

The second triangle has a base of $3$ units (from $-3$ to $0$) and a height of $1$ unit. The area of this triangle is $\frac{1}{2} \times 3 \times 1 = 1.5$

step 5

Since the graph of $g(t)$ is above the $t$-axis in this interval, the areas are positive. Therefore, the total area under $g(t)$ from $-5$ to $0$ is $1 + 1.5 = 2.5$

step 6

Thus, $h(0) = 2.5$

Answer

$h(0) = 2.5$

Key Concept

Definite Integral

Explanation

The definite integral of a function over an interval can be interpreted as the area under the curve of the function within that interval.

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