I dont understand this problem

Question

How is the answer to this not -6? Please solve this without U substitution because I need to know what I did wrong to get a negative answer. I did interval (B) - (A) as you should

$$\int_{-21}^0 \dfrac {\mathrm dx}{\sqrt{~4-x~}} ~$$

Problem-image

Answer 1

How is it not -6?

The negative sign changes the direction of the integral's bounds.

Lets do it in two substitution steps. Use the substitution $v:=4-x$ and then $u:=\surd v$ .

$$\begin{align}\int_{-21}^0 \dfrac{\mathrm d x}{\sqrt{~4-x~}} &= \int_{4-21}^{4-0}-\dfrac{\mathrm d v}{\surd v}&~:~&\mathrm d v =-\mathrm d x\\[2ex]&=\int_{4}^{25}\dfrac{\mathrm d v}{\surd v}\\[2ex]&=\int_{\surd 4}^{\surd 25} 2~\mathrm d u &~:~&\mathrm du=\dfrac{1}{2\surd v}\mathrm d v \\[2ex]&= 2~(5-2)\\[2ex]&=6 \end{align}$$

Or "without" substitution

$$\begin{align}\int_{-21}^0 \dfrac{\mathrm d x}{\sqrt{~4-x~}} &= \int_{x=21}^{x=0} -2\mathrm d \surd(4-x)\\[2ex] &= -2(\surd(4-0)-\surd(4+21))\\[2ex]&=6\end{align}$$