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#### Overview of this book

Are you looking to start developing artificial intelligence applications? Do you need a refresher on key mathematical concepts? Full of engaging practical exercises, The Statistics and Calculus with Python Workshop will show you how to apply your understanding of advanced mathematics in the context of Python. The book begins by giving you a high-level overview of the libraries you'll use while performing statistics with Python. As you progress, you'll perform various mathematical tasks using the Python programming language, such as solving algebraic functions with Python starting with basic functions, and then working through transformations and solving equations. Later chapters in the book will cover statistics and calculus concepts and how to use them to solve problems and gain useful insights. Finally, you'll study differential equations with an emphasis on numerical methods and learn about algorithms that directly calculate values of functions. By the end of this book, you’ll have learned how to apply essential statistics and calculus concepts to develop robust Python applications that solve business challenges.
Preface
1. Fundamentals of Python
Free Chapter
2. Python's Main Tools for Statistics
3. Python's Statistical Toolbox
4. Functions and Algebra with Python
5. More Mathematics with Python
6. Matrices and Markov Chains with Python
7. Doing Basic Statistics with Python
8. Foundational Probability Concepts and Their Applications
9. Intermediate Statistics with Python
10. Foundational Calculus with Python
11. More Calculus with Python
12. Intermediate Calculus with Python

# Using Integrals to Solve Applied Problems

If a curve is rotated about the x or y axis or a line parallel to one of the axes, to form a 3D object, we can calculate the volume of this solid by using the tools of integration. For example, let's say the parabola y = x2 is rotated around its axis of symmetry to form a paraboloid, as in Figure 10.16:

Figure 10.16: A parabola rotated about the z axis

We can find the volume by adding up all the slices of the paraboloid as you go up the solid. Just as before, when we were using rectangles in two dimensions, now we're using cylinders in three dimensions. In Figure 10.16, the slices are going up the figure and not to the right, so we can flip it in our heads and redefine the curve y = x2 as y = sqrt(x).

Now the radius of each cylinder is the y value, and let's say we're going from x = 0 to x = 1:

Figure 10.17: Flipping the paraboloid on its side

The endpoints are still 0...