Function $g$ is graphed.

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

Question

Function $g$ is graphed.

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

Accepted Answer

∻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 $\frac{1}{2} imes 2 imes 2 = 2$. 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 $\frac{1}{2} imes 2 imes 2 = 2$. 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.⚹◊What is the total area under the curve of the function $g(x)$ from $0⍭ Generate me a similar question◊

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