The volume of the cut-off sphere using three planes using the R-language (integral)

Question

I want to calculate the volume (V) of the part of a sphere, which is the result of the intersection of the sphere with three fingers (x = 0, y = 0 and z = 1.5). I use R language and this is my code. I tried 2 different methods using Cartesian and polar coordinates. Both of them give negative answers.

##  Compute the Volume between 3 planes x=0, y=0 and z=1.5 and a sphere
library("pracma", lib.loc="~/R/win-library/3.1")
f <- function(x, y)  (sqrt(4 -x^2 - y^2) - 1.5 ) # here the function(x,y)  is subtracted with -1.5 which represents the plane z=1.5 
xmin <- 0
xmax <- 2
ymin <- 0 
ymax <- function(x) (sqrt(4 - x^2))
I <- integral2(f, xmin, xmax, ymin, ymax)
I$Q   # Integral over sector is not so exact

# Exact Volume from AutoCAD V=0.3600



##  Volume of the sphere: use polar coordinates
f0 <- function(x, y) (sqrt(4 - x^2 - y^2)-1.5)  # for x^2 + y^2 <= 4 the f(x,y) means z changes between zmin=1 and zmax= sqrt(4-x^2-y^2)
fp <- function(t, r) r * f0(r*cos(t), r*sin(t))
quad2d(fp, 0, pi/2, 0, 2, n = 101)  # -0.523597

The correct answer is: V = 0.3600. Can someone give me a hint please?

Greetings

Answer 1

X-Y , f(x,y)-1.5 , . z=1.5 sqrt(7/4) ( Pythagoras), , :

library(pracma)
f <- function(x, y)  (sqrt(4 -x^2 - y^2) - 1.5 ) # here the function(x,y)  is subtracted with -1.5 which represents the plane z=1.5 
xmin <- 0
xmax <- sqrt(7/4)
ymin <- 0 
ymax <- function(x) (sqrt(7/4 - x^2))
I <- integral2(f, xmin, xmax, ymin, ymax)
I$Q   # Integral over sector is not so exact
# [1] 0.3599741

, .

Answer 2

r = 2 x = 1, y = 1 z = 1.

##  Compute the Volume between 3 planes x=1.0, y=1.0 and z=1.0 and a sphere
library(pracma)
f <- function(x, y)  (sqrt(4 -x^2 - y^2) - 1 ) # here the function(x,y) is  subtracted with -1.5 which represents the plane z=1.5 
xmin <- 1
xmax <- sqrt(2)
ymin <- 1 
ymax <- function(x) (sqrt(3 - x^2))
I <- integral2(f, xmin, xmax, ymin, ymax)
I$Q   # 
# [1] 0.01520549
# Exact Volume from AutoCAD: 0.0152