Questions: Find the total area between the x-axis and the curve f(x)=-x^2+4 between x=-1 and x=3. 29/3 units ^2 28/3 units ^2 34/3 units ^2 20/3 units ^2

Transcript text: Find the total area between the $x$-axis and the curve $f(x)=-x^{2}+4$ between $x=-1$ and $x=3$. $\frac{29}{3}$ units $^{2}$ $\frac{28}{3}$ units $^{2}$ $\frac{34}{3}$ units $^{2}$ $\frac{20}{3}$ units $^{2}$

Solution

Solution Steps

Step 1: Define the Function

The function given is $ f(x) = -x^2 + 4 $. This is a downward-opening parabola.

Step 2: Set the Interval

We need to find the area between the curve and the $x$-axis from $x = -1$ to $x = 3$.

Step 3: Calculate the Definite Integral

To find the total area, we calculate the definite integral of $ f(x) $ over the interval $[-1, 3]$: \[ \text{Area} = \int_{-1}^{3} (-x^2 + 4) \, dx \] Evaluating this integral gives us: \[ \text{Area} = \frac{20}{3} \]