Questions: Find the area between the curves. y=x^2-30, y=5-2x The area between the curves is □ (Type an integer or decimal rounded to the nearest tenth as needed.)

Find the area between the curves.
y=x^2-30, y=5-2x

The area between the curves is □
(Type an integer or decimal rounded to the nearest tenth as needed.)

Transcript text: Find the area between the curves. \[ y=x^{2}-30, y=5-2 x \] The area between the curves is $\square$ (Type an integer or decimal rounded to the nearest tenth as needed.)

Solution

Solution Steps

To find the area between the curves $ y = x^2 - 30 $ and $ y = 5 - 2x $, we need to:

Determine the points of intersection by solving $ x^2 - 30 = 5 - 2x $.
Integrate the difference of the functions $ (5 - 2x) - (x^2 - 30) $ between the points of intersection.
Compute the definite integral to find the area.

Step 1: Determine the Points of Intersection

To find the area between the curves $ y = x^2 - 30 $ and $ y = 5 - 2x $, we first need to determine their points of intersection. We solve the equation: \[ x^2 - 30 = 5 - 2x \] Rearranging terms, we get: \[ x^2 + 2x - 35 = 0 \] Solving this quadratic equation, we find the roots: \[ x = -7 \quad \text{and} \quad x = 5 \]

Step 2: Set Up the Integral

Next, we set up the integral to find the area between the curves. The area $ A $ is given by the integral of the difference of the functions from $ x = -7 $ to $ x = 5 $: \[ A = \int_{-7}^{5} \left( (5 - 2x) - (x^2 - 30) \right) \, dx \] Simplifying the integrand: \[ A = \int_{-7}^{5} \left( 35 - 2x - x^2 \right) \, dx \]

Step 3: Compute the Integral

We compute the definite integral: \[ A = \int_{-7}^{5} (35 - 2x - x^2) \, dx \] Evaluating this integral, we get: \[ A = 288 \]