MHB Solving Double Integrals: Order of Integration Explained

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To solve the double integral by changing the order of integration, it is essential to sketch the region of integration first. This visualization helps in determining the correct bounds for the new order. The discussion suggests that the integral in question may be $$\int_{-3}^1\int_{2x}^{3-x^2}xy\,dy\,dx$$. By using horizontal strips instead of vertical ones, one can accurately set the new limits for integration. Understanding this process is crucial for successfully computing the solution.
Estelle
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Hey guys,

need your help and hope someone takes the time:

I need to solve the double integral by changing the order of Integration.

View attachment 4966

I would really appreciate if you could illustrate the way of how to compute the solution.

Best
Estelle :)
 

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Hello and welcome to MHB, Estelle! :D

I have moved this thread, since it is a problem more likely found in the 3rd semester of an elementary calculus course.

Are you certain the problem isn't actually:

$$\int_{-3}^1\int_{2x}^{3-x^2}xy\,dy\,dx$$?
 
Assuming that what Mark has posted is correct, to reverse the order of integration you need to SKETCH the region of integration, and then figure out your bounds by using horizontal strips instead of vertical ones...
 

Thread 'Proving that convexity implies second order derivative being positive'

There are probably loads of proofs of this online, but I do not want to cheat. Here is my attempt: Convexity says that $$f(\lambda a + (1-\lambda)b) \leq \lambda f(a) + (1-\lambda) f(b)$$ $$f(b + \lambda(a-b)) \leq f(b) + \lambda (f(a) - f(b))$$ We know from the intermediate value theorem that there exists a ##c \in (b,a)## such that $$\frac{f(a) - f(b)}{a-b} = f'(c).$$ Hence $$f(b + \lambda(a-b)) \leq f(b) + \lambda (a - b) f'(c))$$ $$\frac{f(b + \lambda(a-b)) - f(b)}{\lambda(a-b)}...
View full post »

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