https://file.aiquickdraw.com/mathbot/file/17233872333240r4vy4sn.jpg

The problem asks to find the area between the curves \( y = x^2 - 8x + 4 \) and \( y = -3 \).

To find this area, we first determine the intersection points of the curves by setting \( x^2 - 8x + 4 = -3 \). Solving this equation:

\[
x^2 - 8x + 7 = 0
\]

Factoring this gives:

\[
(x - 7)(x - 1) = 0
\]

Thus, \( x = 7 \) and \( x = 1 \) are the points of intersection. These will be the bounds for the integration.

To find the area between the curves, we compute the integral of the upper curve minus the lower curve from \( x = 1 \) to \( x = 7 \):

\[
\text{Area} = \int_{1}^{7} [(x^2 - 8x + 4) - (-3)] \, dx = \int_{1}^{7} (x^2 - 8x + 7) \, dx
\]

Calculating this integral:

\[
\int (x^2 - 8x + 7) \, dx = \frac{x^3}{3} - 4x^2 + 7x
\]

Evaluating from \( x = 1 \) to \( x = 7 \):

\[
\left[\frac{x^3}{3} - 4x^2 + 7x \right]_{1}^{7}
\]

Let's compute this value.The computed area between the curves from \( x = 1 \) to \( x = 7 \) is \( -36 \). Since the area cannot be negative and it indicates we calculated the integral in the standard direction (left to right), the absolute value of this result, \( 36 \), is the actual area between the curves.

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 area calculated?
2. Do you need a step-by-step breakdown of how the antiderivative was computed?
3. Would you like to know more about how to factor quadratic equations?
4. Are you interested in solving similar problems with different quadratic functions?
5. Would you like to explore the impact of changing the bounds of integration?
6. How about learning more about integration techniques for finding areas between curves?
7. Do you need help with any other topics in calculus?
8. Can I assist you with another math problem or concept?

**Tip:** When calculating areas between curves, always ensure the upper function is subtracted by the lower function to avoid negative area results.

Learn how to calculate the area between curves x^2 - 8x + 4 and y = -3 using calculus. This problem involves finding intersection points, setting up integrals, and computing definite integrals. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation