Question

Riemann sums for a semicircle Let $f(x)=\sqrt{1-x^2}.$ a. Show that the graph of $f$ is the upper half of a circle of radius 1 centered at the origin. b. Estimate the area between the graph of $f$ and the $x$ -axis on the interval [-1,1] using a midpoint Riemann sum with $n=25$. c. Repeat part (b) using $n=75$ rectangles. d. What happens to the midpoint Riemann sums on [-1,1] as $n\to \mathrm{\infty }?$

Short answer

Answer: As the number of rectangles (n) used in the midpoint Riemann sum increases, the series approaches the actual area under the curve. In the limit as $n$ approaches infinity, the series converges to $\frac{\pi }{2}$, which is half of the area of a circle with radius 1.

Step-by-step solution

Find the equation of the semicircle

To find the equation of the semicircle, we need to consider the equation of a circle. A circle with radius 1 and centered at the origin has the equation $x^2+y^2=1$. Solving for $y$, we get $y^2=1-x^2$ and $y=\pm \sqrt{1-x^2}$. However, the given function $f(x)=\sqrt{1-x^2}$ only represents the upper half of the circle, so the graph is indeed the upper semicircle.

Estimate the area with n = 25.

To estimate the area under the curve using a midpoint Riemann sum, we need to divide the interval $[-1,1]$ into 25 equal parts. The width of each rectangle, $\mathrm{\Delta }x$, will be $\frac{1-(-1)}{25}=\frac{2}{25}$. The midpoints of each interval will be the values for which we evaluate the function. If we denote the midpoint of the $i$-th interval as $x_i^{\ast }$, we can calculate the area with the formula: $$A\approx \sum _{i=1}^{25}f(x_i^{\ast })\mathrm{\Delta }x$$Let's calculate this sum.

Estimate the area with n = 75.

Repeating the process with n = 75 rectangles, we adjust our $\mathrm{\Delta }x$ and midpoints: $\mathrm{\Delta }x=\frac{2}{75}$, and the formula for the area becomes:$$A\approx \sum _{i=1}^{75}f(x_i^{\ast })\mathrm{\Delta }x$$Calculate this sum as well.

Analyze as n approaches infinity.

As the number of rectangles used in the midpoint Riemann sum (n) increases, the approximation of the area becomes increasingly accurate. In the limit as $n\to \mathrm{\infty }$, the midpoint Riemann sum converges to the actual area under the curve. This can be described as:$$\underset{n\to \mathrm{\infty }}{lim}(\sum _{i=1}^{n}f(x_i^{\ast })\mathrm{\Delta }x)={\int }_{-1}^{1}f(x)dx$$The actual area under the curve is equal to half of the area of a circle with radius 1, which is $\frac{1}{2}\pi (1{)}^2=\frac{\pi }{2}$. The series of midpoint Riemann sums will approach this value as $n$ approaches infinity.

Key concepts

Semicircle

A semicircle is essentially half of a full circle. In the case of a function like $f(x)=\sqrt{1-x^2}$, it describes the upper half of a circle with a radius of 1. This stems from the equation of a circle, $x^2+y^2=1$, where the circle is centered at the origin and has a radius of 1.
The function only covers the part of the circle where $y\ge 0$, which captures the top half. Hence, the graph is called a semicircle.
Everything from its symmetry about the y-axis to its smooth continuous curve reflects properties inherent to semicircles.
Understanding semicircles is fundamental in recognizing how the function behaves and what the graph looks like, which is crucial for further concepts like integrals and Riemann sums.

Midpoint Approximation

The midpoint approximation is a method of estimating the area under a curve or function over a certain interval. In our scenario, we are using it to approximate the area beneath the semicircle's upper half on the interval $[-1,1]$.
We split this interval into $n$ equal parts or rectangles. The midpoint of each subinterval is then used to estimate the height of each rectangle, hence the term "midpoint."
The more rectangles we use (higher $n$), the narrower they become. This is beneficial because it provides a more accurate estimation of the area.
For example, with $n=25$, each rectangle's width $\mathrm{\Delta }x$ is $\frac{2}{25}$, and with $n=75$, $\mathrm{\Delta }x$ becomes $\frac{2}{75}$.
The sum of the areas of these rectangles gives us an approximation of the total area under the curve.

Integrals

Integrals are mathematical tools used for finding areas under curves, among other things. The process of integration involves summing an infinite number of infinitesimally small quantities, which fits precisely with the concept of calculating areas.
In connection with Riemann sums, integrals represent the limit of the sum of areas of approximating rectangles as the number of rectangles $n$ approaches infinity. This ensures that the integral provides the exact area under the curve.
The integral of our semicircle's function on $[-1,1]$ helps in determining the precise area, which should be $\frac{\pi }{2}$, as it is half the area of a circle with radius 1.
This precise relationship between Riemann sums and integrals underscores why both concepts are essential in calculus.

Convergence

Convergence is a term used to describe when a sequence or series approaches a specific value as it progresses. For Riemann sums, it refers to how our approximation becomes more accurate as the number of rectangles, $n$, increases.
When considering the midpoint Riemann sums for our semicircle from $[-1,1]$, as $n\to \mathrm{\infty }$, the series of sums becomes increasingly close to the exact area under the curve, thanks to the nature of convergence.
This is encapsulated in the relationship $\underset{n\to \mathrm{\infty }}{lim}(\sum _{i=1}^{n}f(x_i^{\ast })\mathrm{\Delta }x)={\int }_{-1}^{1}f(x)dx$.
Convergence ensures that, with enough rectangles, our estimated area becomes as good as the real value. It's a vital concept, as it provides us the confidence in mathematical practices like integration.