It might seem artificial to write a sum of the products, like 1×100 or 4×1, but that's just what the expanded form is. Centimeters — divide the volume value by 28 , ⁣ 316.847 28,\!316.847 28 , 316.847 (which is 30.4 8 3 30.48 Use Math Input above or enter your integral calculator queries using plain English. To avoid ambiguous queries, make sure to use parentheses where necessary. Here are some examples illustrating how to ask for an integral using plain English. The expanded form is a way to write a number as a sum, each summand corresponding to one of the number's digits. In our case, the sum would be: Still, we might wish to decompose it even further. After all, we wanted to see the digits themselves (i.e., as one-digit numbers) and not some " complicated" expression like 0.07. Therefore, we can also write:

As other people (who are probably real mathematicians) have implied, as you get further on in your mathematical career, the less useful is to think of mathematical constructs being real. Instead it's helpful to think of them being useful (or in some cases elegant but of no practical use - although that's largely what people thought of number theory, before the invention of public key cryptography). The indefinite integral of , denoted , is defined to be the antiderivative of . In other words, the derivative of is . Since the derivative of a constant is 0, indefinite integrals are defined only up to an arbitrary constant. For example,, since the derivative of is . The definite integral of from to , denoted , is defined to be the signed area between and the axis, from to . Convert the volume directly to cubic feet unit. You may find this method easier, as you only need to divide or multiply once: The sum we got can encourage us to go even further! After all, we can get 100, 10, 1, 0.1, and 0.01 by raising the number 10 to integer powers: to the power 2, 1, 0, -1, and -2, respectively. In other words, we can also write: which is the number we had initially but with the point two places to the right. This movement by 2 is shown by the power in the standard form exponents.

Order Approval

As you can see, we had five digits, so we got five terms. What is more, consecutive digits appear in consecutive summands; we simply add a few zeros in the correct places to make it all jump to the right spot when we add it all up. Non-Americans often refer to the standard form in math in connection with a very different topic. To be precise, they understand it as the basic way of writing numbers (with decimals) using the decimal base (as opposed to, say, the binary base), which we can decompose into terms representing the consecutive digits. Now, this looks even worse than the previous example; it doesn't have commas in between! Thankfully, there are tools - like our standard form calculator - to make our lives easier. So, what is the standard form of the above numbers? After converting the units, you'll have all of the dimensions in feet, so a simple multiplication will give us the result in cubic feet. We said that the number b should be between 1 and 10. This means that, for example, 1.36 × 10⁷ or 9.81 × 10⁻²³ are in standard form, but 13.1 × 10¹² isn't because 13.1 is bigger than 10. We could, however, convert it to standard form by saying that:

To return to your original question though: imagine you have a number line and want to double a number, x. You get an imaginary rope, cut it to length x then lay it out from 0 to x then from x to 2x. This is easily generalized to multiplying by any natural number, a. Conversely, if we divide the initial number by 10, which is equal to multiplying it by 1/10 = 10⁻¹, we'll get Don't ask us how they found the mass of the Earth, as there isn't any scale big enough to weigh the entire planet. As for the circumference, talk to Eratosthenes. Now that we've seen how to write a number in standard form, it's time to convince you that it's a useful thing to do. Of course, we know that you're most probably learning all of this for the pure pleasure of grasping yet another part of theoretical mathematics, but it doesn't hurt to take a look at physics or chemistry from time to time. You know, those two minor branches of mathematics.To divide by two you measure out a length of rope, then grab both ends and you have a length of x/2. You can generalise to divide by any natural number, b. What are integrals? Integration is an important tool in calculus that can give an antiderivative or represent area under a curve. Now, this is more like it! We don't know about you, but for us, short is beautiful, in mathematics at least.

We've spent quite some time together with the standard form calculator, enough to know that we can't leave the answer like this. We haven't learned how to write a number in standard form for nothing. But there's more! We have multiplication and division in the formula, and the standard form exponents make these two operations very easy to calculate. By the well-known, well-remembered, and totally not forgotten the moment the test was over formulas, multiplying two powers with the same base is the same as adding the exponents, while dividing corresponds to subtracting them. In other words, if we separate the 10s to some powers from the other numbers, we'll get:Both types of integrals are tied together by the fundamental theorem of calculus. This states that if is continuous on and is its continuous indefinite integral, then . This means . Sometimes an approximation to a definite integral is desired. A common way to do so is to place thin rectangles under the curve and add the signed areas together. Wolfram|Alpha can solve a broad range of integrals How Wolfram|Alpha calculates integrals Wolfram|Alpha computes integrals differently than people. It calls Mathematica's Integrate function, which represents a huge amount of mathematical and computational research. Integrate does not do integrals the way people do. Instead, it uses powerful, general algorithms that often involve very sophisticated math. There are a couple of approaches that it most commonly takes. One involves working out the general form for an integral, then differentiating this form and solving equations to match undetermined symbolic parameters. Even for quite simple integrands, the equations generated in this way can be highly complex and require Mathematica's strong algebraic computation capabilities to solve. Another approach that Mathematica uses in working out integrals is to convert them to generalized hypergeometric functions, then use collections of relations about these highly general mathematical functions. and the circumference is... actually, the 40,075 km doesn't look that bad, does it? Well, we could use a length converter and change it to 4.0075 × 10⁴ km, but is it better that way? If we needed to change it to millimeters, then maybe it'd be a better idea, but the kilometer form seems perfectly usable. Suppose that you've taken up astronomy recently and would like to know the gravitational force acting between the Earth and the Moon. For the calculations, we need the masses of the two objects (denote the Earth's by M₁ and the Moon's by M₂) and the distance between them (denoted by R). We have: