finding the area between 2 curves by karen bendel multiple - choice question we need three separate…

finding the area between 2 curves by karen bendel multiple - choice question we need three separate integrals here. the first one goes from -3 to -2, the second one -2 to 2, and the third one from 2 to 3. which is the correct set up for the third integral? ∫₂³x² - (8 - x²)dx ∫₂³8 - x² - x²dx rewatch submit

finding the area between 2 curves by karen bendel multiple - choice question we need three separate integrals here. the first one goes from -3 to -2, the second one -2 to 2, and the third one from 2 to 3. which is the correct set up for the third integral? ∫₂³x² - (8 - x²)dx ∫₂³8 - x² - x²dx rewatch submit

Answer

Explanation:

Step1: Recall area - between - curves formula

The area between two curves $y = f(x)$ and $y = g(x)$ from $x=a$ to $x = b$ is given by $A=\int_{a}^{b}|f(x)-g(x)|dx$. We need to determine which function is on top and which is on the bottom for $x\in[2,3]$.

Step2: Analyze the curves

Assume the two curves are $y = x^{2}$ and $y=8 - x^{2}$. We can find which one is greater for $x\in[2,3]$. Let's take a value $x = 2.5$ (in the interval $[2,3]$). For $y_1=x^{2}$, when $x = 2.5$, $y_1=(2.5)^{2}=6.25$. For $y_2=8 - x^{2}$, when $x = 2.5$, $y_2=8-(2.5)^{2}=8 - 6.25 = 1.75$. So for $x\in[2,3]$, $y=x^{2}$ is above $y = 8 - x^{2}$.

Step3: Set up the integral

The area between the two curves from $x = 2$ to $x=3$ is $\int_{2}^{3}[x^{2}-(8 - x^{2})]dx$.

Answer:

$\int_{2}^{3}[x^{2}-(8 - x^{2})]dx$