Questions: What is the area between the curve g(x)=3 x^2+2 and the x-axis from x=-2 to x=0?

Transcript text: What is the area between the curve $g(x)=3 x^{2}+2$ and the $x$-axis from $x=-2$ to $x=0$?

Solution

Solution Steps

To find the area between the curve $ g(x) = 3x^2 + 2 $ and the $ x $-axis from $ x = -2 $ to $ x = 0 $, we need to compute the definite integral of $ g(x) $ over the interval $[-2, 0]$.

Step 1: Define the Function

The function given is $ g(x) = 3x^2 + 2 $.

Step 2: Set the Integration Limits

We need to find the area between the curve and the $ x $-axis from $ x = -2 $ to $ x = 0 $.

Step 3: Compute the Definite Integral

Calculate the definite integral of $ g(x) $ over the interval $[-2, 0]$: \[ \int_{-2}^{0} (3x^2 + 2) \, dx \]

Step 4: Evaluate the Integral

The integral evaluates to: \[ \left[ x^3 + 2x \right]_{-2}^{0} = (0^3 + 2 \cdot 0) - ((-2)^3 + 2 \cdot (-2)) \] \[ = 0 - (-8 - 4) = 12 \]

Final Answer

The area between the curve and the $ x $-axis is $\boxed{12}$.

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