{"id":22452,"date":"2023-02-27T01:49:02","date_gmt":"2023-02-27T01:49:02","guid":{"rendered":"https:\/\/www.booksofall.com\/?post_type=product&p=22452"},"modified":"2023-02-27T02:27:19","modified_gmt":"2023-02-27T02:27:19","slug":"apex-calculus2","status":"publish","type":"product","link":"https:\/\/www.booksofall.com\/pt\/apex-calculus2\/","title":{"rendered":"APEX Calculus 2"},"content":{"rendered":"
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Read More:\u00a0Calculus 1\u00a0<\/a>& Calculus 3<\/a><\/strong><\/p>\n

5: Integration<\/h2>\n

We have spent considerable me considering the derivatives of a function and their applications<\/a>. In the following chapters, we are going to star ng thinking in \u201cthe other direction.\u201d That is, given a function f<\/i>(x<\/i>), we are going to consider functions F<\/i>(x<\/i>) such that\u00a0F \u2032<\/i>(x<\/i>) =\u00a0f<\/i>(x<\/i>). There are numerous reasons this will prove to be useful: these functions will help us compute area, volume, mass, force, pressure, work, and much more.<\/p>\n

5.1 An derivatives and Indefinite Integration <\/b><\/p>\n

Given a function y\u00a0<\/i>=\u00a0f<\/i>(x<\/i>), a\u00a0differential equation <\/i>is one that incorporates\u00a0y<\/i>,\u00a0x<\/i>, and the derivatives of y<\/i>. For instance, a simple differential equation is:<\/p>\n

y \u2032\u00a0<\/i>= 2x.\u00a0<\/i><\/p>\n

Solving a differential equation amounts to finding a function y\u00a0<\/i>that satisfies the given equation. Take a moment and consider that equation; can you find a function y\u00a0<\/i>such that\u00a0y \u2032\u00a0<\/i>= 2x<\/i>?<\/p>\n

Can you find another?<\/p>\n

And yet another?<\/p>\n

Hopefully one was able to come up with at least one solution: y\u00a0<\/i>=\u00a0x<\/i>2. \u201cFinding another\u201d may have seemed impossible un l one realizes that a function like y\u00a0<\/i>=\u00a0x<\/i>2 + 1 also has a derivative of 2x<\/i>. Once that discovery is made, finding \u201cyet another\u201d is not difficult; the function y\u00a0<\/i>=\u00a0x<\/i>2 + 123,\u00a0<\/i>456,\u00a0<\/i>789 also has a derivative of 2x<\/i>. The differential equation y \u2032\u00a0<\/i>= 2x\u00a0<\/i>has many solutions. This leads us to some definitions.<\/p>\n

Definition 5.1.1
\nAn derivatives and Indefinite Integrals <\/b><\/em><\/p>\n

Let a function f(x) be given. An\u00a0an derivative <\/b>of\u00a0f(x) is a function F(x) such that\u00a0F \u2032(x) =\u00a0f(x).<\/em>
\nThe set of all an derivatives of f(x) is the\u00a0indefinite integral of\u00a0<\/b>f, denoted by \u222b f(x)\u00a0dx.<\/em><\/p>\n

Make a note about our definition: we refer to an\u00a0<\/i>an derivative of f<\/i>, as op- posed to\u00a0the\u00a0<\/i>an derivative of f<\/i>, since there is\u00a0always\u00a0<\/i>an infinite number of them.<\/p>\n<\/div>\n<\/div>\n

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We often use upper-case letters<\/a> to denote an derivatives. Knowing one an derivative of <\/span>f\u00a0<\/i>allows us to find infinitely more, simply by<\/span><\/div>\n<\/div>\n
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adding a constant. Not only does this give us more\u00a0<\/i>an derivatives, it gives us all\u00a0<\/i>of them.<\/p>\n

Theorem 5.1.1
\nAn derivative Forms <\/b><\/p>\n

Let\u00a0F<\/i>(x<\/i>) and\u00a0G<\/i>(x<\/i>) be an derivatives<\/a> of f<\/i>(x<\/i>) on an interval\u00a0I<\/i>. Then there exists a constant\u00a0C\u00a0<\/i>such that, on\u00a0I<\/i>,
\nG<\/i>(x<\/i>) =\u00a0F<\/i>(x<\/i>) +\u00a0C.<\/i><\/p>\n

Given a function f\u00a0<\/i>defined on an interval\u00a0I\u00a0<\/i>and one of its an derivatives F<\/i>, we know\u00a0all\u00a0<\/i>an derivatives of f\u00a0<\/i>on\u00a0I\u00a0<\/i>have the form\u00a0F<\/i>(x<\/i>) +\u00a0C\u00a0<\/i>for some constant\u00a0C<\/i>. Using Definition 5.1.1, we can say that
\n\u222bf<\/i>(x<\/i>)\u00a0dx\u00a0<\/i>=\u00a0F<\/i>(x<\/i>) +\u00a0C.<\/i><\/p>\n

Let\u2019s analyze this indefinite integral nota on.<\/p>\n

\"\"<\/p>\n

(Figure 5.1.1: Understanding the indefinite integral nota on)<\/p>\n

Figure 5.1.1 shows the typical nota on of the indefinite integral<\/a>. The integration symbol<\/a>, \u222b , is in reality an \u201celongated S,\u201d representing \u201ctake the sum.\u201d We will later see how\u00a0sums\u00a0<\/i>and\u00a0an derivatives <\/i>are related.<\/p>\n

The function we want to find an an derivative of is called the integrand<\/i>. It contains the differential of the variable we are integrating with respect to. The \u222b symbol and the differential dx\u00a0<\/i>are not \u201cbookends\u201d with a function sandwiched in between; rather, the symbol \u222b means \u201cfind all an derivatives of what follows,\u201d and the function f<\/i>(x<\/i>) and\u00a0dx\u00a0<\/i>are multiplied together; the dx\u00a0<\/i>does not \u201cjust sit there.\u201d<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"