Definite Integrals

Learn definite integrals, signed area and geometric area.

Lesson 2 of 32 parts Practice Flashcards

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Definite Integrals

Objective: Explain definite integrals, apply the stated method, and check its conditions.

A definite integral is the limit of suitable Riemann sums when that limit exists. It represents signed accumulation across an interval. For a positive constant function, the integral equals rectangle height times interval width; it is a number, not a function family. A definite integral is a numerical accumulation, not an indefinite family with +C.

Formulas

$∫ₐᵇ k dx=k(b−a)$

Worked examples

Try itEvaluate ∫₀^7 6 dx.Show answer

Step 1: A constant positive function forms a rectangle. Step 2: Height=6, width=7. Step 3: The definite integral is 42.

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Definite Integrals

Objective: Explain definite integrals, apply the stated method, and check its conditions.

Formulas

$∫ₐᵇ k dx=k(b−a)$

Worked examples

Try itEvaluate ∫₀^7 6 dx.Show answer

Step 1: A constant positive function forms a rectangle. Step 2: Height=6, width=7. Step 3: The definite integral is 42.

Signed Area and Geometric Area

Objective: Explain signed area and geometric area, apply the stated method, and check its conditions.

A definite integral assigns negative contributions where the integrand is negative. Geometric area is nonnegative and corresponds to integrating the absolute value when measuring area against an axis. Positive and negative contributions may cancel in net accumulation. Net signed area can differ from total geometric area.

Formulas

Geometric area=∫|f(x)|dx

Worked examples

Try itEvaluate ∫₀^7 (−6) dx.Show answer

Step 1: The function lies below the x-axis. Step 2: The integral counts signed area, (−6)·7. Step 3: The integral is -42; geometric area is 42.

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