# Why aren't the integrals $\int_{-1}^2 \sqrt{t^2}\sqrt{9t^2+4} \,dt$ and $\int_{-1}^2 t\sqrt{9t^2+4} \,dt$ equal?

I'm a bit confused by the two integrals below. Why aren't they equal? What am I missing?

$$\int_{-1}^2 \sqrt{t^2}\sqrt{9t^2+4} \,dt = 10,51$$

$$\int_{-1}^2 t\sqrt{9t^2+4} \,dt = 7,63$$

• You are assuming $\sqrt{t^2}=t$. – kimchi lover May 11 at 19:16
• Because $t\ne\sqrt{t^2}$ in general. – Lord Shark the Unknown May 11 at 19:16
• Always use : $\sqrt{t^2}=|t|$, to be correct and consistent. – Dr Zafar Ahmed DSc May 13 at 5:25

Since $$\sqrt{t^2}=|t|$$ they are not equal.
The integrand of the second integral is negative on $$(-1,0)$$, while the first is not.