{"id":31,"date":"2017-01-26T20:12:18","date_gmt":"2017-01-27T03:12:18","guid":{"rendered":"https:\/\/blogs.ubc.ca\/cuishan\/?p=31"},"modified":"2017-01-26T20:12:18","modified_gmt":"2017-01-27T03:12:18","slug":"assignment-3","status":"publish","type":"post","link":"https:\/\/blogs.ubc.ca\/cuishan\/2017\/01\/26\/assignment-3\/","title":{"rendered":"Assignment 3"},"content":{"rendered":"

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Integration<\/span><\/i> and<\/span> antidifferentiation<\/span><\/i> are commonly conflated since it\u2019s generally misunderstood that they are both the inverse of differentiation. However, they are totally different in their exact definitions, despite that they are indeed connected in some way. Integration is the process of calculating integrals, i.e. the area under a fixed curve, which is a number. Whereas antidifferentiation is the process of finding the antiderivatives of a function, which are a number of functions. Comparing these two vague definitions, it can be assumed that antidifferentiation is not equal to integration. <\/span><\/p>\n

What is antidifferentiation ?<\/span><\/p>\n

Antidifferentiation \u00a0can be understood as the inverse of differentiation. <\/span><\/p>\n

For a function f(x), the <\/span>antiderivative <\/b>(primitive function) of this function, <\/span>f(x)dx<\/span>, are a group of \u00a0functions F(x)+C that satisfies the derivative of this group functions is f(x). <\/span>f(x)dx=F(x)+C<\/span>. So the <\/span>antiderivative <\/b>of f(x) is not unique. <\/span><\/p>\n

example:<\/span><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/b><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>f(x)=1 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0F(x)=x+c. (F(x) could be F(x)=x+1, F(x)=x+2\u2026 )<\/span><\/p>\n

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What is integration?<\/span><\/p>\n

For integration, integral can be defined by <\/span>Riemann Sum<\/span><\/i>. <\/span><\/p>\n

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let f(t) dt defined on <\/span>a closed interval [<\/span>a<\/span><\/i>, <\/span>b<\/span><\/i>]<\/span> and divide <\/span>\u00a0[<\/span>a<\/span><\/i>, <\/span>b<\/span><\/i>] into n sunbintervals<\/span><\/p>\n

with the same <\/span>t<\/span> value, choose a sample ponit <\/span>t<\/span>i<\/span>*<\/span> within <\/span>[t<\/span>i-1<\/span>,<\/span>t<\/span>i<\/span>]<\/span>, so that the area of one rectangles is<\/span>f(<\/span>t<\/span>i<\/span>*<\/span>)<\/span>t<\/span>, so the area of these rectangles is \u00a0<\/span>n<\/span>inf<\/span>i=1<\/span>n<\/span>f(<\/span>t<\/span>i<\/span>*<\/span>)<\/span>t<\/span>.<\/span><\/p>\n

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Given a continuous function f(x) and an interval <\/span>[<\/span>a<\/span><\/i>, <\/span>b<\/span><\/i>], the meaning of <\/span>\u00a0integral<\/b> a<\/span>b<\/span>f(x) dx <\/span>is the area of curve trapezoid formed by function f(x), x=a , x=b and x-axis in x-y plane (as illustrated in this picture). \u00a0So integration is a certain real number.<\/span><\/p>\n

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What is the connection between integration and antidifferentiation ?<\/span><\/p>\n

Integration and antidifferentiation are correlated by<\/span> Fundamental Theorem of Calculus<\/span><\/i> (FTC) since an integral can be evaluated by using antiderivatives. As \u00a0FTC2 said, if G(x) is an antiderivative of f(x), then we have,<\/span><\/p>\n

This means to evaluate the integral of continuous function f(x) defined on [a, b], we just need to take antiderivative of the function. In this way, after finding an antiderivative of a function, we can simply plug in x=b and x=a, and subtract these two values. Hence it is where they are connected that antidifferentiation can be used in calculating integrals.<\/span><\/p>\n

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Why students commonly mistake \u201cIntegration\u201d& \u201cAntiderivation\u201d?<\/span><\/p>\n

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