# Solving Ap Calc AB Problems: Area, Volume, & Cross Sections

• kreil
In summary, the conversation involves finding the area and volume of a region bounded by two given graphs. The area is found by calculating the integral of the difference between the two graphs, and the volume is found by revolving the region around the x-axis and calculating the integral of the squared difference between the two graphs. Finally, the volume of a solid with the region as its base and square cross sections parallel to the x-axis is found by calculating the integral of the squared difference between the two graphs.
kreil
Gold Member
I have been having trouble with this kind of problem lately and I need to know if what I have done here is right (calculator problem but I have not evaluated anything yet):

Let R be the region bounded by the y-axis and the graphs of

$$y=\frac{x^3}{1+x^2}$$ and

$$y=4-2x$$

a) Find the area of R
b) Find the volume of the solid generated when R is revolved about the x-axis
c)The region R is the base of a solid. For this solid, each cross section parallel to the x-axis is a square. Find the volume of this solid.

a)
a=point of intersection of the two graphs

$$\int_0^a(4-2x-\frac{x^3}{1+x^2})dx$$

b)
$${\pi}\int_0^a(4-2x)^2-(\frac{x^3}{1+x^2})^2dx$$

c)
$$\pi \int_0^a(4-2x-\frac{x^3}{1+x^2})^2dx$$

Last edited:
there shouldn't be a $\pi$ in front of the integral on letter c.

everything else is fine

First of all, it is great that you are seeking help and checking your work. It is important to make sure your solutions are correct when solving calculus problems.

For part a), finding the area of R, you correctly set up the integral using the bounds of 0 and a, and the two given functions. However, there is a small error in your integrand. Remember that when finding the area between two curves, the top function should be subtracted by the bottom function. So the correct integral should be:

\int_0^a(4-2x)-(\frac{x^3}{1+x^2})dx

For part b), finding the volume of the solid generated when R is revolved about the x-axis, you correctly set up the integral using the bounds of 0 and a, and the two given functions. However, the integrand should be the volume of a disc, which is {\pi}r^2, where r is the distance from the function to the axis of rotation. So the correct integral should be:

{\pi}\int_0^a[(4-2x)-(\frac{x^3}{1+x^2})]^2dx

For part c), finding the volume of the solid where each cross section parallel to the x-axis is a square, you correctly set up the integral using the bounds of 0 and a, and the two given functions. However, the integrand should be the area of a square, which is (side length)^2. In this case, the side length is the difference between the two functions. So the correct integral should be:

\int_0^a[(4-2x)-(\frac{x^3}{1+x^2})]^2dx

Overall, your setup and approach to solving these problems is correct. Just make sure to pay attention to the integrand and the difference between finding area and volume. Keep practicing and you will continue to improve. Good luck!

## 1. What are the key concepts to keep in mind when solving AP Calculus AB problems involving area, volume, and cross sections?

The key concepts to keep in mind when solving these types of problems are understanding the definitions of area and volume, knowing how to calculate the area and volume of basic shapes, understanding how to set up and solve integrals for more complex shapes, and being able to use cross sections to find the volume of irregular shapes.

## 2. How can I approach a problem involving finding the volume of a solid with known cross sections?

When solving a problem involving finding the volume of a solid with known cross sections, it is important to first visualize the solid and its cross sections. Then, set up a definite integral using the given cross sections and the variable of integration. Finally, use the Fundamental Theorem of Calculus to evaluate the integral and find the volume.

## 3. What are some common mistakes to avoid when solving AP Calculus AB problems related to area, volume, and cross sections?

Some common mistakes to avoid include not carefully reading and understanding the given problem, using incorrect formulas for calculating area and volume, forgetting to include units in the final answer, and making errors in setting up and evaluating integrals.

## 4. How can I check my answers for AP Calculus AB problems involving area, volume, and cross sections?

You can check your answers by using a graphing calculator or online tool to graph the given function and the calculated area or volume. You can also use the answer key or solutions manual to compare your answer to the correct answer. Additionally, you can try solving the problem using a different method to see if you get the same result.

## 5. How can I improve my problem-solving skills for AP Calculus AB problems related to area, volume, and cross sections?

To improve your problem-solving skills, practice solving a variety of problems involving different types of shapes, cross sections, and methods of integration. Also, make sure to understand the underlying concepts and formulas rather than just memorizing them. Seek help from a tutor or teacher if you are struggling with a particular concept or problem type.

• Introductory Physics Homework Help
Replies
25
Views
2K
• Introductory Physics Homework Help
Replies
64
Views
2K
• Introductory Physics Homework Help
Replies
14
Views
2K
• Introductory Physics Homework Help
Replies
21
Views
1K
• Introductory Physics Homework Help
Replies
6
Views
997
• Introductory Physics Homework Help
Replies
7
Views
1K
• Introductory Physics Homework Help
Replies
36
Views
517
• Introductory Physics Homework Help
Replies
52
Views
2K
• Introductory Physics Homework Help
Replies
4
Views
698
• Introductory Physics Homework Help
Replies
3
Views
772