Question

Math

Posted 3 months ago

```
Function $g$ is graphed.
Find $\int_{0}^{4} g(x) d x$. Give an exact answer as a multiple of $\pi$.
$\int_{0}^{4} g(x) d x=$
$\square$
```

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Answer from Sia

Posted 3 months ago

Solution by Steps

step 1

To find the integral $\int_{0}^{4} g(x) \, dx$, we need to analyze the graph of the function $g$ from $x = 0$ to $x = 4$

step 2

From the graph, we observe that $g(x)$ decreases from $(0, 0)$ to $(2, -2)$ and then increases from $(2, -2)$ to $(4, 0)$

step 3

The area under the curve from $x = 0$ to $x = 2$ is a triangle with base $2$ and height $2$. The area of this triangle is $\frac{1}{2} \times 2 \times 2 = 2$. Since the function is below the x-axis, this area is negative: $-2$

step 4

The area under the curve from $x = 2$ to $x = 4$ is another triangle with base $2$ and height $2$. The area of this triangle is $\frac{1}{2} \times 2 \times 2 = 2$. Since the function is above the x-axis, this area is positive: $2$

step 5

Adding these areas together, we get the total area under the curve from $x = 0$ to $x = 4$: $-2 + 2 = 0$

Answer

$0$

Key Concept

Area under the curve

Explanation

The integral of a function over an interval can be found by calculating the areas of the regions under the curve, taking into account whether the function is above or below the x-axis.

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