What is the area between the curves 𝑦=−𝑥2+4y=−x2+4 and 𝑦=−3y=−3?

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

\[
-x^2 + 4 = -3 \implies -x^2 = -7 \implies x^2 = 7
\]

So, \( x = \pm \sqrt{7} \) are the intersection points.

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

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

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 \( y = -x^2 + 4 \) and \( y = -3 \) from \( x = -\sqrt{7} \) to \( x = \sqrt{7} \) is approximately \( 24.69 \) square units.

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

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**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.

Learn how to calculate the area between the curves y = -x^2 + 4 and y = -3 using definite integrals. Detailed step-by-step solution provided. Suitable for advanced high school or college students.

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