Ytukyg

I'd like to understand the proof of the following Proposition 6.7 (Book: Gruber.P Convex and Discrete Geometry, page 103)

$$text{Proposition: Let} C in mathcal{C}(mathbb{E}^{d-1}) text{and embed} mathbb{E}^{d-1} text{into} mathbb{E}^{d} text{as usual} (text{first} d-1 text{ coordinates}). Then \W_i(C) = frac{i kappa_i}{d kappa_{i-1}}omega_{i-1}(C) text{for} i=1,...,d. $$

Where $W_i$ is the $d$-dimensional Quermass-Integral and $omega_i$ is the $d-1$-dimensional Quermass-Integral. $v(cdot)$ is the volume and $kappa_i$ is the $i$-dimensional volume of the Ball $B^{i}$

Proof: Let u=(0,...,0,1). Then

$$sum_{i=0}^{d}binom{d}{i} W_i(C)lambda^{i}=V(C+lambda B^d)overset{1}{=} int limits_{-lambda}^{lambda} v((C+lambda B^d)cap(mathbb{E}^d +tu))dt \ overset{2}{=}int limits_{- lambda}^{lambda} v(C+(lambda^{2}-t^2)^{1/2}B^{d-1})dt = ... =sum_{i=0}^{d-1}binom{d-1}{i}omega_i(C) int limits_{- lambda}^{lambda}(lambda^{2}-t^2)^{i/2}dt\ overset{3}{=}sum_{i=0}^{d-1}binom{d-1}{i}omega_i(C)frac{kappa_{i+1}}{kappa_{i}}lambda^{i+1}$$

1) I don't understand how to get this equality. I know it is a Fubini argument but im not able to calculate it. I tryed to do it with the indicator function but i don't get the bounds.

2) Can someone explain me where $(lambda^{2}-t^2)^{1/2}$ comes form? I thought when i intersect $C$ with $mathbb{E}^d+tu$ i get $C$ und if i intersect $lambda B^d$ with $mathbb{E}^d+tu$ i should get $lambda B^{d-1}$ if the intersection is a great circle. If it is not a great circle than it has a elliptic form ? How can i visualize this?

3) Can someone help me to integrate $int limits_{- lambda}^{lambda} (lambda^{2}-t^2)^{i/2}dt$ or has a reference where i can see how it is done ?

Thank you in advance.