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.