All Questions
Tagged with bounds-of-integration substitution
5
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How is this property of definite integral derived?
The property:
$$
\int_a^b f(x) \, dx=\int_a^b f(a+b-x) \, dx
$$
Derivation given in my textbook:
Let $t = a+b-x$. Then $dt = -d x$. When $x=a, t=b$ and when $x=b, t=a$. Therefore,
$$
\begin{aligned}
\...
- 33
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Question on the bounds of definite integration during a substitution
Apologies if this question is rather elementary. I seem to still misunderstand a few things about how bounds change during substitutions still.
I was taught in calc II that to perform a substitution, ...
- 3,515
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Find $\int_{\left(C\right)}xy{\rm d}x+y^{2}{\rm d}y$ with $\left(C\right)$ bound by $y\geq 0,x^{2}+y^{2}=4\left({\rm clockwise}\right).$
Prob. Find $\int_{\left ( C \right )}xy{\rm d}x+ y^{2}{\rm d}y$ with $\left ( C \right )$ closed by the path $y\geq 0, x^{2}+ y^{2}= 4\left ( {\rm clockwise} \right ).$
My attempt: $\int_{\left ( C \...
user822157
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How to evaluate $\iint_R \sin(\frac{y-x}{y+x})dydx$ with Jacobian substitution?
I want to calculate this integral with substitution $x=u+v , \ y=u-v$:
$$\iint_R \sin\left(\frac{y-x}{y+x}\right)dydx$$
$$R:= \{(x,y):x+y≤\pi, y≥0,x≥0\}$$
but I don't know how to set new bounds for $...
- 301
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votes
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Is it necessary to write limits for a substituted integral?
To solve the following integral, one can use u-substitution:
$$\int_2^3 \frac{9}{\sqrt[4]{x-2}} \,dx,$$
With $u = \sqrt[4]{x-2}$, our bounds become 0 and 1 respectively. Thus, we end up with the ...
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