query: Why is Integral from $a$ to $b$ equal to negative integral from $b$ to $a$
positive_passages.docid: 27754
positive_passages.text: Is integration from $a$ to $b$ same or $b$ to $a$ or is negative?
negative_passages.docid: 48920
negative_passages.text: Mean value theorem for integrals: how does the sign matter?
negative_passages.docid: 97184
negative_passages.text: Show: $ f(a) = a,\ f(b) = b \implies \int_a^b \left[ f(x) + f^{-1}(x) \right] \, \mathrm{d}x = b^2 - a^2 $
negative_passages.docid: 80733
negative_passages.text: integration of function equals zero
negative_passages.docid: 102825
negative_passages.text: Why change the sign of the integral when switching the limits of integration?
negative_passages.docid: 6892
negative_passages.text: Why is it not true that $\int_0^{\pi} \sin(x)\; dx = 0$?
negative_passages.docid: 15947
negative_passages.text: Validity of Substitutions in Integrals
negative_passages.docid: 76238
negative_passages.text: Why is the relative velocity of B with respect to A negative of the relative velocity of A with respect to B?
negative_passages.docid: 95145
negative_passages.text: Does $\exp(2ir\pi)$ equal $1$? What's wrong?
negative_passages.docid: 63251
negative_passages.text: Evaluation of $\int^1_0\frac{x^b-x^a}{\ln x} d x=\ln{{\frac{1+b}{1+a}}}$
negative_passages.docid: 66404
negative_passages.text: Prove that:$\int_{a}^{b} f(x) dx = b \cdot f(b) - a \cdot f(a) - \int_{f(a)}^{f(b)} f^{-1}(x) dx$
negative_passages.docid: 141852
negative_passages.text: Prove that $2\int_a^b \int_a^x f(x)f(y) \, dy \, dx = \left[ \int_a^b f(x) \, dx \right]^2$
negative_passages.docid: 73489
negative_passages.text: How is the Integral of $\int_a^bf(x)~dx=\int_a^bf(a+b-x)~dx$
negative_passages.docid: 36697
negative_passages.text: Proving that $\int_{-b}^{-a}f(-x)dx=\int_{a}^{b}f(x)dx$
negative_passages.docid: 125764
negative_passages.text: Under what conditions does Mathematica give a natural antiderivative when integrating?
negative_passages.docid: 30694
negative_passages.text: Show that there exist $[a,b]\subset [0,1]$, such that $\int_{a}^{b}f(x)dx$ = $\int_{a}^{b}g(x)dx$ = $\frac{1}{2}$
negative_passages.docid: 33542
negative_passages.text: When is $A- B = B- A$?