Can you help me with this double integral?

Question

$\int_{0}^{1} \int_{3 y}^{3} e^{x^{2}} d x d y$ $int_0^1int_(3y)^3e^(x^2)dxdy$

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Answer 1 · 2018-06-11T16:31:38

Similar to your previous question, change the order of integration.

We have $x$ $x$ bounded by $x = 3 y$ $x=3y$ (which is also $y = \frac{1}{3} x$ $y = 1/3x$ ) and by $x = 3$ $x=3$ .

Graph those lines:

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We need $3 y \leq x \leq 3$ $3y <=x<=3$ and also $0 \leq y \leq 1$ $0<=y<=1$ , so the region over which we are integrating looks like:

enter image source here

To change the order of integration, we'll need limits of integrating for

$int int e^(x^2) \ dy \ dx$

$y$ goes from $0$ to the line $y=1/3x$ and $x$ goes from $0$ to $3$ .

So we need to evaluate

$int_0^3 int_0^(1/3x) e^(x^2) \ dy \ dx$