Pedantically, Goodham (குதம் in Tamil script) refers specifically to the sheath inside the anal orifice & does not denote the anus or the buttocks. Strictly, the anus is distinguished by its own term, namely 'Goodoushtham'. However, 'Goodham' is often vulgarly & inaccurately extended to also mean "arse-hole" & even the whole "butt", which, in case of Dravidian women, often attains a pleasingly large size.
Since the same one Tamil letter is used to depict 'k' & 'g', & another is used for both 't' & 'd', variant dialectal pronunciations & spellings include: Kootham, Koottam, Kutam, Kotham, Kutham, Koothi, Gūdham, Gudam, Guddam, & Godham.
The word has entered Indo-Aryan languages, such as Magadhi or Pali: "gudaṁ, The anus. ... Ab.274." ('A Dictionary of the Pali Language' Robert Caesar Childers. London: Tru:bner, 1872-5, p.150) Of course, Sanskrit, one of the most highly Dravidianized Indo-Aryan tongues, has long adopted this term: "The lower end of the large intestine is called gudam (rectum) & this ends in gudoushtha (anus). The interior of the rectum is provided with three spiral grooves. These ring-like muscles lie a finger & a half apart from one another & are respectively known as pravāhini, visarjani & samvarani ..." ('Indian system of medicine'. History of Science & Technology in India, vol 4. O P Jaggi. Delhi: Atma Ram, 1986, p.114)
2) "kutam ... s. the backside; 2. the anus" ('Tamil & English Dictionary'. J P Fabricius. 4th ed. Tranquebar: Evangelical Lutheran Mission Pub House, 1972, p.259. cf tamilenglishdictionary.com)
3) The variants Koothi, Kootham & Kotham are derived from Gudham, as Darx notes: "Koothi, Kotham (derived from gudham) = anus" (darx 19 Jun 2006, comment on 'Get Your "Kundis" to the D.C. Meetup, 3/25' Anna. Sepia Mutiny (14 Mar 2006), sepiamutiny.com/sepia/archives/003147.html). This etymology is also provided in the Tamil Lexicon: "kutam, n. < guda. Anus." ('Tamil Lexicon'. Univ Of Madras. Madras: Univ Of Madras, 1924-36, p.990)
4) In Brahminical Kannadiga, one has the Sanskrit derivative: "guda. The anus; the rectum" ('A Kannada-English dictionary', Rev F Kittel, Mangalore: Basel Mission Book & Tract Depository, 1894, p.553)