One of the most popular queries on Wolfram|Alpha is for definite integrals. So we’re especially excited to announce that Step-by-step solutions for these are now available! The general method used to find the steps for definite integrals is to tap into the already existing “Show steps” functionality for indefinite integration, and then to use the fundamental theorem of calculus. More »

Ever since we added Pokémon data to Wolfram|Alpha last fall, it’s been interesting to see the recurring traffic spikes as word spreads and the linkbacks on the interwebs continue to grow. You also wanted to know who are the most-searched Pokémon, so we thought we’d share an update on what our site data says so far. More »

Did you know that October 11 is World Egg Day?

Like most people, you probably go to the grocery store and eventually end up in the dairy aisle, where, unless you’re a vegetarian or vegan, you probably pick out a dozen eggs and place them into your cart without a second thought. They’re pretty much a staple food—from savory breakfasts to the sweet wonders of baking. More »

We here at Wolfram are, by and large, a bunch of nerds. This shouldn’t be that surprising, especially after looking at the people and fictional characters we’ve turned into mathematical curves on Wolfram|Alpha. Our curves of internet memes, cartoon and video game characters, celebrities, and mathematical formulas have been incredibly popular. As many of our fellow nerds get ready to go back to school, we’re celebrating nerddom and showing our appreciation for Wolfram’s users—by letting one of *you* decide the next curve to be featured in Wolfram|Alpha. More »

I love dogs; they are the best. I find that they are suitable not only as companions, but as friends and confidants. That said, as much as I might anthropomorphize them, I do genuinely wish I could see the world in their eyes. Now, with Wolfram|Alpha, I can—and so can you. More »

Throughout the history of physics, scientists have postulated laws and theories about the nature of the world around them. Some were proven false, while others have grown to be the basis of entire fields of study. One such field is classical mechanics, which describes the area of physics most familiar to us, that of the motion of macroscopic objects, from baseballs to planets and traveling along hills to falling from space. As one of the oldest subjects in science, the work here serves as a basis for less familiar areas such as relativity and quantum mechanics. More »

When I was younger, I held the naive and incorrect view that mathematics was divorced from the arts. As I’ve gotten older, I’ve become more aware of not only how mathematics is the foundation for any of the hard sciences, but also how it is intrinsically linked to essentially any form of creativity. Certainly users of our Wolfram Music Theory Course Assistant could have told me that, but I’m not just referring to music. In truth, I’m not even trying to make some highbrow appeal to abstract art, either, although I happen to rather like that sort of thing. What I’m trying to say is that mathematical equations can make pretty pictures.

So, you forgot your anniversary was coming up, and now you have to decide what you’re going to get your loved one. Wolfram|Alpha can now help point you in the right direction. The stereotypical anniversary gift for a man to give his wife is often thought to be jewelry, but you would be surprised to know that many traditional and modern wedding gifts have nothing to do with jewelry.

What do the following two math problems have in common?

- If I have 12 apples, and Jane has 7, and then Jane gives 2 apples to me, how many more apples do I have than Jane?
- (12 + 2) – (7 – 2)

Answer: two things, actually. More »

Step-by-step solutions, one of the most popular features for mathematics in Wolfram|Alpha, has just received a dramatic expansion in its functionality! With our new interface, you now have the ability to walk through all of our Step-by-step solutions at your own pace, revealing only one step at a time. Some of our programs will offer to guide you with hints when walking through solutions. And for common math problems, we can even show multiple ways to find the solutions. More »

Like many people, I went to see *Total Recall* recently. Much of the story for this science fiction thriller (based loosely on a short story by Philip K. Dick) concerns “The Fall.” As explained in the introduction for the movie, “The Fall” is a tunnel bored through the center of the Earth that connects the fictional United Federation of Britain and The Colony, which is located in present day Australia. More »

One of the features of calculus is the ability to determine the arc length or surface area of a curve or surface. An arc length is the length of the curve if it were “rectified,” or pulled out into a straight line. You can also think of it as the distance you would travel if you went from one point to another along a curve, rather than directly along a straight line between the points.

To see why this is useful, think of how much cable you would need to hang a suspension bridge. The shape in which a cable hangs by itself is called a “catenary,” but with a flat weight like a roadway hanging from it, it takes the shape of a more familiar curve: a parabola.

Remember that New Year’s resolution you made to lose weight this year? If you’re one of the many people around the world who pledged to get healthy and finally lose that weight, Wolfram|Alpha is here to help! Even with January behind us, there is still plenty of time to get back on track in 2012.

Studies throughout the decades have shown that regular diet and exercise is the number-one way to get healthy. Wolfram|Alpha can offer you a variety of different ways to start, track, and maintain your new healthy lifestyle.

Query how to lose weight, and Wolfram|Alpha will bring up a formula where you can enter all the information needed, including your intended physical activity level, to figure out how many calories you should be eating every day in order to reach your target body weight:

What do your alarm clock, thermostat, coffeemaker, car radio, anti-lock brakes—and almost every other electrical and mechanical device you encounter in your daily life—all have in common? They are all examples of “control systems,” one of the most ubiquitous yet unseen modern technologies. A control system is any system or device that controls or regulates the behavior of another system. Using various kinds of sensors and actuators, these systems automatically control most common appliances, industrial processes, and even your body’s own biological processes!

Take your home’s humble thermostat. The temperature of your home depends on many factors, especially how long and how recently the home’s furnace was on. With a thermostat installed, the reverse is also true: the state of the furnace depends on the temperature of the house (it comes on if the temperature is too low, and turns off if the temperature is too high). There is a closed loop of causation formed between the home’s temperature and the state of the furnace. By design, the thermostat creates a kind of closed loop called a “negative feedback loop,” which tends to stabilize the temperature around a desired value. Most control systems are like this: sensors feed information back into the system, which is then used to decide on an action. More »

At one time or another, we’ve all looked at a jet flying high overhead and thought “I wonder where they’re headed?” Actually answering that question probably seemed impossible before—but if you’re a user in the United States, Wolfram|Alpha can now help you answer that question and many more interesting queries about commercial and other flights.

Try the simple query “flights overhead” and you’ll get information on aircraft that should be visible to you, assuming a clear sky and unobstructed view. If you’re on a location-aware mobile device, the results should be based on your precise latitude and longitude—otherwise, Wolfram|Alpha will use the best available location information from your browser. Also note that hovering over an individual plane in the sky map will produce a tooltip with the airline and flight number:

In my last blog post on plotting functionality in Wolfram|Alpha, we looked at 2D and 3D Cartesian plotting. In this post, we will look at 2D polar and parametric plotting.

For those of you unfamiliar with polar plots, a point on a plane in polar coordinates is located by determining an angle θ and a radius *r*. For example, the Cartesian point (*x*, *y*) = (1, 1) has the polar coordinates (*r*, θ) = (√2,π/4). The following diagram illustrates the relationship between Cartesian and polar plots.

To generate a polar plot, we need to specify a function that, given an angle θ, returns a radius *r* that is a function *r*(θ). Making a polar plot in Wolfram|Alpha is very easy; for example, we can plot Archimedes’ spiral. More »

Need a tutor for solving equations? Solving equations is just one of hundreds of mathematical tasks that can be done using Wolfram|Alpha. Wolfram|Alpha can solve equations from middle school level all the way through college level and beyond. So next time you are stumped on an equation, consult Wolfram|Alpha for a little help.

Let’s start with the simpler stuff. Wolfram|Alpha can easily solve linear and quadratic equations, and even allows you to view a step-by-step solution of each problem.

What if the roots of the equation are complex? No worries; Wolfram|Alpha has no trouble solving equations over the complex plane.

Wolfram|Alpha can also solve cubic and quartic equations in terms of radicals.

Of course, some solutions are too large or cannot be represented in terms of radicals; Wolfram|Alpha will then return numerical solutions with a “More digits” button. More »

*(January 15, 2014 Update: Step-by-step solutions has been updated! Learn more.)*

Have you ever given up working on a math problem because you couldn’t figure out the next step? Wolfram|Alpha can guide you step by step through the process of solving many mathematical problems, from solving a simple quadratic equation to taking the integral of a complex function.

When trying to find the roots of 3*x*^{2}+*x*–7=4*x*, Wolfram|Alpha can break down the steps for you if you click the “Show steps” button in the Result pod.

As you can see, Wolfram|Alpha can find the roots of quadratic equations. Wolfram|Alpha shows how to solve this equation by completing the square and then solving for *x*. Of course, there are other ways to solve this problem! More »