Calculus allows you to measure complex shapes that are normally too difficult. Think, for example, about trying to find out how much water is in a long, oddly shaped lake – it would be impossible to measure each gallon of water separately or use a ruler to measure the shape of the lake. Calculus allows you to study how the edges of the lake change, and use that information to learn how much water is inside. Making geographic models and studying volume is using integration. Integration is the second major branch of calculus. Integration is used to measure the space underneath any line, which allows you to find the area of odd or irregular shapes. Take the equation y=4−x2,{\displaystyle y=4-x^{2},} which looks like an upside-down “U.” You might want to find out how much space is underneath the U, and you can use integration to find it. While this may seem useless, think of the uses in manufacturing – you can make a function that looks like a new part and use integration to find out the area of that part, helping you order the right amount of material. You cannot just integrate an entire function. For example, y=x{\displaystyle y=x} is a diagonal line that goes on forever, and you cannot integrate the whole thing because it would never end. When integrating functions, you need to choose an area, such as all points between x=2{\displaystyle x=2} and x=5.{\displaystyle x=5.} Imagine you have a flat line above a graph, like y=4.{\displaystyle y=4.} To find the area underneath it, you would be finding the area of a rectangle between y=0{\displaystyle y=0} and y=4.{\displaystyle y=4.} This is easy to measure, but it will never work for curvy lines that cannot be turned into rectangles easily. If you zoom in very close to a curve, it looks flat. This happens every day – you cannot see the curve of the earth because we are so close to its surface. Integration makes an infinite number of little rectangles under a curve that are so small they are basically flat, which allows you to measure them. Add all of these together to get the area under a curve. Imagine you are adding together a lot of little slices under the graph, and the width of each slice is ‘’almost’’ zero. Integrals come with 4 parts. A typical integral looks like this:∫f(x)dx{\displaystyle \int f(x)\mathrm {d} x}  The first symbol, ∫,{\displaystyle \int ,} is the symbol for integration (it is actually an elongated S). The second part, f(x),{\displaystyle f(x),} is your function. When it is inside the integral, it is called the integrand.  Finally, the dx{\displaystyle \mathrm {d} x} at the end tells you what variable you are integrating with respect to. Because the function f(x){\displaystyle f(x)} depends on x,{\displaystyle x,} that is what you should integrate with respect to. Remember, the variable you are integrating is not always going to be x,{\displaystyle x,} so be careful what you write down. . Integration comes in many forms, and you will need to learn a lot of different formulas to integrate every function. However, they all follow the principles outlined above: integration sums up an infinite number of things.  Integrate by substitution. Calculate indefinite integrals. Integrate by parts. This is an ironclad rule of calculus that is so important, it has its own name: the Fundamental Theorem of Calculus. Since integration and differentiation are so closely related, a combination of the two of them can be used to find rate of change, acceleration, speed, location, movement, etc. no matter what information you have. For example, remember that the derivative of speed is acceleration, so you can use speed to find acceleration. But if you only know the acceleration of something (like objects falling due to gravity), you can integrate it to find the speed! Spinning a flat shape around is a way to create 3D solids. Imagine spinning a coin on the table in front of you – notice how it appears to form a sphere as it spins. You can use this concept to find volume in a process known as “volume by rotation.” This lets you find the volume of any solid in the world, as long as you have a function that mirrors it. For example, you can make a function that traces the bottom of a lake, and then use that to find the volume of the lake, or how much water it holds.
اعرف أنك تستخدم التفاضل والتكامل لإيجاد مساحات وأحجام معقدة. اعرف أن التكامل يوجد المساحة أسفل الرسم البياني. اعرف أنك ينبغي أن تختار مساحة لإيجاد تكاملها. تذكر كيفية إيجاد مساحة مستطيل. اعرف أن التكامل يجمع الكثير من المستطيلات الصغيرة لإيجاد المساحة. اعرف كيف تقرأ التكامل وتكتبه. تعلم كيف توجد التكامل. اعرف أن التكامل يعكس التفاضل، والعكس صحيح. اعرف أن التكامل من الممكن أيضًا أن يحسب حجم أجسام ثلاثية الأبعاد.