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 what do we do if the dimensions are expressed in units other than feet — like yards, inches, or even centimeters and meters? Well, you need to convert them. There are two ways of doing this — either you convert the units before the volume calculation or after: 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.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: 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. 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 .

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: 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. F = 0.00000000006674 N·m²/kg² × 5,972,000,000,000,000,000,000,000 kg × 73,480,000,000,000,000,000,000 kg / (384,400,000 m)². 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? 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:What are integrals? Integration is an important tool in calculus that can give an antiderivative or represent area under a curve.

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. This time, we indeed see the digits as the first factors in each multiplication. Moreover, the second factors have a lot in common - they consist of a single 1 with some zeros (possibly none). Before we give some examples of writing numbers in standard form in physics or chemistry, let's recall from the above section the standard form math formula: 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.

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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. 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.

For our non-American friends out there, the standard form is usually quite a different thing. Outside of the USA (especially in the UK), we say that a number is in its standard form if it's a single value that involves no arithmetic operations whatsoever. This notion is connected to the expanded form, and we explain it all in detail in the dedicated section. Also, note how you can switch between the two variants in the advanced section by choosing the appropriate option in the field " Have the calculator use..." 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. 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. For instance, take the number 154.37. It is in its standard form in the decimal base. That means 1 is the hundreds digit, 5 is that of tens, 4 of ones, 3 of tenths, and 7 of hundredths. Having the number written the way it is, makes us see it as a whole, and we don't really think of the individual digits, do we? 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: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. Wolfram|Alpha is a great tool for calculating antiderivatives and definite integrals, double and triple integrals, and improper integrals. The Wolfram|Alpha Integral Calculator also shows plots, alternate forms and other relevant information to enhance your mathematical intuition.