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

Solution Steps

We start by defining the function \( g(x) = 3x^2 + 2 \) which represents the curve we are interested in.

To find the area between the curve and the \( x \)-axis from \( x = -2 \) to \( x = 0 \), we set up the definite integral: \[ \text{Area} = \int_{-2}^{0} g(x) \, dx = \int_{-2}^{0} (3x^2 + 2) \, dx \]

We evaluate the integral: \[ \int (3x^2 + 2) \, dx = x^3 + 2x + C \] Now, we compute the definite integral: \[ \text{Area} = \left[ x^3 + 2x \right]_{-2}^{0} = \left(0^3 + 2 \cdot 0\right) - \left((-2)^3 + 2 \cdot (-2)\right) \] Calculating this gives: \[ \text{Area} = 0 - \left(-8 - 4\right) = 0 - (-12) = 12 \]

Final Answer