find the area between the curves.\ny = x^2 - 8, y = 7 - 2x\n\nthe area between the curves is .\n(type an…

find the area between the curves.\ny = x^2 - 8, y = 7 - 2x\n\nthe area between the curves is .\n(type an integer or decimal rounded to the nearest tenth as needed.)

find the area between the curves.\ny = x^2 - 8, y = 7 - 2x\n\nthe area between the curves is .\n(type an integer or decimal rounded to the nearest tenth as needed.)

Answer

Explanation:

Step1: Find intersection points

Set $x^{2}-8 = 7 - 2x$. Rearrange to $x^{2}+2x - 15=0$. Factor: $(x + 5)(x - 3)=0$. So $x=-5$ and $x = 3$.

Step2: Determine upper - lower curve

For $-5<x<3$, $7 - 2x>x^{2}-8$.

Step3: Use integral formula for area

The area $A=\int_{-5}^{3}[(7 - 2x)-(x^{2}-8)]dx=\int_{-5}^{3}(15 - 2x - x^{2})dx$.

Step4: Integrate

$\int(15 - 2x - x^{2})dx=15x-x^{2}-\frac{1}{3}x^{3}+C$.

Step5: Evaluate definite integral

$A=\left[15x - x^{2}-\frac{1}{3}x^{3}\right]_{-5}^{3}=(15\times3-3^{2}-\frac{1}{3}\times3^{3})-(15\times(-5)-(-5)^{2}-\frac{1}{3}\times(-5)^{3})$. $=(45 - 9 - 9)-(-75-25+\frac{125}{3})$. $=27-(-100+\frac{125}{3})$. $=27 + 100-\frac{125}{3}=\frac{81 + 300 - 125}{3}=\frac{256}{3}\approx85.3$.

Answer:

$85.3$