https://r2.erweima.ai/mathbot/image/17234416341308hs03qs1.jpg

To solve this problem, we need to determine the volume of the solid whose base is bounded by the curves \( p(x) = 2x^2 - 8x - 3.4 \) and \( g(x) = -2x^2 + 4x + 6 \), with cross-sections perpendicular to the x-axis that are semicircles.

### Step 1: Find the intersection points of \( p(x) \) and \( g(x) \)
To find the bounds of integration, we need to find the x-values where \( p(x) = g(x) \). This requires solving the equation:
\[
2x^2 - 8x - 3.4 = -2x^2 + 4x + 6
\]
Combine like terms:
\[
4x^2 - 12x - 9.4 = 0
\]
We can use the quadratic formula to solve for \( x \), where:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \( a = 4 \), \( b = -12 \), and \( c = -9.4 \).

### Step 2: Solve for the area of the cross-sectional semicircles
The area \( A(x) \) of a semicircle is given by:
\[
A(x) = \frac{1}{2} \pi r^2
\]
The radius \( r \) of the semicircle is half the distance between \( p(x) \) and \( g(x) \), so:
\[
r(x) = \frac{p(x) - g(x)}{2}
\]
Thus, the area \( A(x) \) is:
\[
A(x) = \frac{1}{2} \pi \left(\frac{p(x) - g(x)}{2}\right)^2
\]
Simplify this expression:
\[
A(x) = \frac{\pi}{8} \left(p(x) - g(x)\right)^2
\]

### Step 3: Set up and evaluate the integral for volume
The volume \( V \) of the solid is found by integrating \( A(x) \) from the lower intersection point \( x_1 \) to the upper intersection point \( x_2 \):
\[
V = \int_{x_1}^{x_2} A(x) \, dx = \frac{\pi}{8} \int_{x_1}^{x_2} \left(p(x) - g(x)\right)^2 dx
\]
Substitute \( p(x) - g(x) \) and integrate between the limits \( x_1 \) and \( x_2 \) that were found in Step 1.

Let me calculate the exact values of the intersection points and the volume for you.The volume of the solid is approximately \( 304.16 \) cubic units.

Would you like more details on any of the steps, or do you have any other questions?

Here are some related questions you might find helpful:

1. How do you derive the formula for the area of a semicircle?
2. What are the applications of calculating volumes using cross-sectional areas?
3. How would the volume change if the cross-sections were full circles instead of semicircles?
4. Can you solve the problem if the cross-sections were squares instead of semicircles?
5. What happens to the volume if the curves are shifted vertically?
6. How do you solve the quadratic equation using the quadratic formula?
7. What would be the process if the base curves intersect at non-real points?
8. How does the integration process change if the curves are defined piecewise?

**Tip:** When solving for the intersection of two functions, always simplify the resulting equation as much as possible before applying methods like the quadratic formula.

Learn how to calculate the volume of a solid formed between two curves with semicircular cross-sections using integration. Detailed step-by-step solution with intersection points and volume calculation provided.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation