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$$\int_{-1}^{1} \sqrt{1-x^2}dx$$

I let $u = 1-x^2$, $x = (1-u)^{1/2}$

$du = -2x dx$

$$-\frac{1}{2}\int_{0}^{0} \frac{u^{1/2}}{(1-u)^{1/2}} = 0$$ because $$\int_{a}^{a} f(x)dx = 0$$

But it isn't zero. Why?

  • 1
    Under what conditions can you use u substitution? – 2017-02-18
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    It is not zero because the integrand is strictly positive. **Edit:** And continuous. – 2017-02-18
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    f and g' are continuous.. i see now – 2017-02-18
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    Very closely related question (and my answer): http://math.stackexchange.com/questions/1489577/why-is-it-not-true-that-int-0-pi-sinx-dx-0/1489594#1489594 – 2017-02-19
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    @CameronWilliams Duplicates, really. – 2017-02-19

2 Answers 2

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For $x \in [-1,0)$, you can't have $$ u = 1-x^2 \implies x = (1-u)^{1/2} $$ thus your change of variable is not valid over $[-1,0)$.

You would better write by parity $$ \int_{-1}^{1} \sqrt{1-x^2}dx=2\int_0^{1} \sqrt{1-x^2}dx $$ then use the given change of variable.

10

You may only do a u-substitution when it is bijective on your integration domain. You can solve this problem by breaking up your integral into $\int_{-1}^0$ and $\int_0^1,$ and using $x=-(1-u)^{1/2}$on the first integral, and $x=(1-u)^{1/2}$ on the second.