The formula for $\pi$ as an infinite product was derived by English mathematician John Wallis in 1655. This product, called the Wallis Product, appeared in his book Arithmetica Infinitorum.

$$\frac{\pi}{2}=\left(\frac{2 \cdot 2}{1 \cdot 3}\right)\left(\frac{4 \cdot 4}{3 \cdot 5}\right)\left(\frac{6 \cdot 6}{5 \cdot 7}\right) \cdots\left(\frac{(2 n) \cdot(2 n)}{(2 n-1) \cdot(2 n+1)}\right) \cdots$$

In 2015, physicists Carl Hagen and Tamar Friedmann (also a mathematician) stumbled upon a connection between quantum mechanics and the Wallis Product when they applied the variational principle to higher energy states of the hydrogen atom. This principle was previously used only on the ground energy state. The Wallis Product appeared naturally in the midst of their calculations involving gamma functions.

Quantum mechanics is the study of matter and light on the atomic and subatomic scale.

Consider Wallis's method of finding a formula for $\pi$. Let

$$I(n)=\int_0^{\pi / 2} \sin ^n x d x$$

From Wallis's Formulas, $I(n)=\left(\frac{1}{2}\right)\left(\frac{3}{4}\right)\left(\frac{5}{6}\right) \cdots\left(\frac{n-1}{n}\right)\left(\frac{\pi}{2}\right), n$ is even $(n \geq 2)$

or

$I(n)=\left(\frac{2}{3}\right)\left(\frac{4}{5}\right)\left(\frac{6}{7}\right) \cdots\left(\frac{n-1}{n}\right), n$ is odd $(n \geq 3)$

Find $I(n)$ for $n=2,3,4$, and 5 . What do you observe?

Consider the shaded region between the graph of $y=\sin x$, where $0 \leq x \leq \pi$, and the line $y=c$, where $0 \leq c \leq 1$, as shown in the figure. A solid is formed by revolving the region about the line $y=c$. (a) For what value of $c$ does the solid have minimum volume? (b) For what value of $c$ does the solid have maximum volume?

Find the indefinite integral. ∫ √(16 - 4x²) dx

Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. 4 / (x² - 8x)

Evaluate the integrals

$$\int_{-1}^1\left(1-x^2\right) d x \text { and } \int_{-1}^1\left(1-x^2\right)^2 d x .$$

Approximate the definite integral using the Trapezoidal Rule and Simpson’s Rule with n=4. Compare these results with the approximation of the integral using a graphing utility. $\int_{0}^{4} \sqrt{x} e^{x} d x$

Describe how you would find each integral using (i) integration by parts and (ii) substitution. Perform each integration. Are the results equivalent?

$$\int 2 x \sqrt{2 x-3} d x$$

Find the indefinite integral. Do not integrate. ∫ (9+x²)^-2 dx

Approximate the definite integral using the Trapezoidal Rule and Simpson’s Rule with n=4. Compare these results with the approximation of the integral using a graphing utility. $\int_{0}^{4} \sqrt{x} e^{x} dx$

Find the indefinite integral.

$$\int \sqrt{16-4 x^2} d x$$

Use partial fractions to find the indefinite integral. ∫ [(x²+5) / (x³ - x² + x + 3) dx]

Approximate the definite integral using the Trapezoidal Rule and Simpson’s Rule with n = 4. Compare these results with the approximation of the integral using a graphing utility.

$$\int_0^4 \sqrt{x} e^x d x$$

Use Wallis’s Formulas to evaluate the integral. $\int_{0}^{\pi / 2} \sin ^{10} x dx$

Find the indefinite integral.

$$\int\left[z^2+\frac{1}{(1-z)^6}\right] d z$$