The value of ∫ 4 12 6 x 3 d x \displaystyle {\int_{{{4}}}^{{{12}}}}{6}{x}^{{3}}\ {\left.{d}{x}\right.} ∫ 4 12 6 x 3 d x lim n → ∞ ∑ k = 1 n 6 c k 3 Δ x \displaystyle \lim_{{{n}\to\infty}}{\sum_{{{k}={1}}}^{{n}}}{6}{{c}_{{k}}^{{3}}}\ \Delta{x} n → ∞ lim k = 1 ∑ n 6 c k 3 Δ x = lim n → ∞ ∑ k = 1 n ( A Δ x + B Δ x 2 + C Δ x 3 + D Δ x 4 ) \displaystyle =\lim_{{{n}\to\infty}}{\sum_{{{k}={1}}}^{{n}}}{\left({A}\Delta{x}+{B}\Delta{x}^{{2}}+{C}\Delta{x}^{{3}}+{D}\Delta{x}^{{4}}\right)} = n → ∞ lim k = 1 ∑ n ( A Δ x + B Δ x 2 + C Δ x 3 + D Δ x 4 )
lim n → ∞ ∑ k = 1 n A Δ x = \displaystyle \lim_{{{n}\to\infty}}{\sum_{{{k}={1}}}^{{n}}}{A}\Delta{x}= n → ∞ lim k = 1 ∑ n A Δ x = Preview Question 6 Part 1 of 4
lim n → ∞ ∑ k = 1 n B Δ x 2 = \displaystyle \lim_{{{n}\to\infty}}{\sum_{{{k}={1}}}^{{n}}}{B}\Delta{x}^{{2}}= n → ∞ lim k = 1 ∑ n B Δ x 2 = Preview Question 6 Part 2 of 4
lim n → ∞ ∑ k = 1 n C Δ x 3 = \displaystyle \lim_{{{n}\to\infty}}{\sum_{{{k}={1}}}^{{n}}}{C}\Delta{x}^{{3}}= n → ∞ lim k = 1 ∑ n C Δ x 3 = Preview Question 6 Part 3 of 4
lim n → ∞ ∑ k = 1 n D Δ x 4 = \displaystyle \lim_{{{n}\to\infty}}{\sum_{{{k}={1}}}^{{n}}}{D}\Delta{x}^{{4}}= n → ∞ lim k = 1 ∑ n D Δ x 4 = Preview Question 6 Part 4 of 4