Math Problem Statement

What is the area between the curves π‘¦=βˆ’π‘₯2+4y=βˆ’x2+4 and π‘¦=βˆ’3y=βˆ’3?

Solution

To find the area between the curves y=βˆ’x2+4y = -x^2 + 4 and y=βˆ’3y = -3, we first need to identify the points of intersection. Setting βˆ’x2+4=βˆ’3-x^2 + 4 = -3, we solve for xx:

βˆ’x2+4=βˆ’3β€…β€ŠβŸΉβ€…β€Šβˆ’x2=βˆ’7β€…β€ŠβŸΉβ€…β€Šx2=7-x^2 + 4 = -3 \implies -x^2 = -7 \implies x^2 = 7

So, x=Β±7x = \pm \sqrt{7} are the intersection points.

The area between the curves is computed by integrating the difference between the upper curve and the lower curve from x=βˆ’7x = -\sqrt{7} to x=7x = \sqrt{7}. In this case, the upper curve is y=βˆ’x2+4y = -x^2 + 4 and the lower curve is y=βˆ’3y = -3. The integral becomes:

Area=βˆ«βˆ’77[(βˆ’x2+4)βˆ’(βˆ’3)] dx=βˆ«βˆ’77(βˆ’x2+7) dx\text{Area} = \int_{-\sqrt{7}}^{\sqrt{7}} [(-x^2 + 4) - (-3)] \, dx = \int_{-\sqrt{7}}^{\sqrt{7}} (-x^2 + 7) \, dx

Let's compute this integral.It appears I forgot to import the function for the square root. Let me correct this and compute the area again.The area between the curves y=βˆ’x2+4y = -x^2 + 4 and y=βˆ’3y = -3 from x=βˆ’7x = -\sqrt{7} to x=7x = \sqrt{7} is approximately 24.6924.69 square units.

Would you like more details on this solution or have any questions? Here are some follow-up questions and a tip:

  1. Would you like to see a graph of these functions and the calculated area?
  2. Do you need a step-by-step breakdown of how the integral was calculated?
  3. Are you interested in learning how changing the coefficients in the quadratic equation affects the area?
  4. Would you like to solve problems involving other shapes defined by quadratic equations?
  5. How about exploring the use of definite integrals to find volumes?
  6. Do you need help with integration techniques for more complex functions?
  7. Can I assist you with another calculus topic?
  8. Is there another math problem or concept you'd like help with?

Tip: Remember, for symmetric functions about the y-axis, you can often simplify calculations by integrating from 0 to a positive bound and then doubling the result.

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Math Problem Analysis

Mathematical Concepts

Calculus
Definite Integrals
Quadratic Equations

Formulas

Area between curves formula
Integral of a polynomial

Theorems

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Suitable Grade Level

Advanced High School or College