Example 28: ∫ from −1 to 2 of |x³ − x| dx = 11/4उदाहरण 28: −1 से 2 तक ∫ |x³ − x| dx = 11/4Example 28: −1 se 2 tak ∫ |x³ − x| dx = 11/4

NCERT Example 28 · Class 12 Maths · Chapter 7 · interactive diagram

Parts and what they mean

• y = |x^3 - x| (y = |x^3 - x|) — The modulus flips the part below the axis (between 0 and 1) upward.
• y = x^3 - x (dashed) (y = x^3 - x (टूटी रेखा)) — The original cubic; it is negative between 0 and 1.
• Roots x = −1, 0, 1 (मूल x = −1, 0, 1) — x^3 - x = x(x - 1)(x + 1) changes sign here, so we split the integral at 0 and 1 (P₂).
• Piece on [−1, 0] = 1/4 ([−1, 0] वाला भाग = 1/4) — \int_{-1}^{0}(x^3 - x)\,dx = \dfrac{1}{4}
• Piece on [0, 1] = 1/4 ([0, 1] वाला भाग = 1/4) — -\int_0^1 (x^3 - x)\,dx = \dfrac{1}{4} (minus, because the cubic is negative here).
• Piece on [1, 2] = 9/4 ([1, 2] वाला भाग = 9/4) — \int_1^2 (x^3 - x)\,dx = \dfrac{9}{4}. Total = \dfrac{11}{4}.
• Upper limit x = 2 (उच्च सीमा x = 2) — At x = 2, |x^3 - x| = 6.
• x-axis (x-अक्ष) — The horizontal number line; inputs x are read here.
• y-axis (y-अक्ष) — The vertical number line; outputs y = f(x) are read here.
• Origin O(0, 0) (मूल बिंदु O(0, 0)) — Where the two axes cross.

Original diagram drawn by Simple Learning Hub, based on NCERT Example 28. Report a mistake