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Find: (i). Why is it that for one variable it is not usually necessary to understand how the...

Question:

Why is it that for one variable it is not usually necessary to understand how the interval changes under the change of variable? Why is it important for functions of several variables to understand how the region changes under the change of variable?

Change Of Variables

Change of variable is used to simplify an integral. For an integral {eq}\int_{a}^{b} f(x) \,dx {/eq}, substituting {eq}x=h(t) {/eq}, we get:

{eq}\int_{a}^{b} f(x) \,dx = \int_{h(a)}^{h(b)} f[h(t)]\cdot h'(t)\,dt {/eq}

Answer and Explanation:

In the case of one variable, straight lines map to straight lines when performing a change of variables. But in case of more variables, it is highly probable that the shape of the region changes. For example, in the case of 2 variables, a square may map to a circle. Hence it is important to understand how the region changes.


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How to Solve Integrals Using Substitution

from Math 104: Calculus

Chapter 13 / Lesson 5
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