--- title: "The Definite Integral and the Area Under a Curve" description: "Estimate the area under a curve with rectangles, then evaluate it exactly by substituting the limits into the antiderivative. Areas below the axis come out negative and have to be handled separately." canonical: https://lightmysky.com/learn/mathematics/the-definite-integral-and-the-area-under-a-curve-mt_S2IO2S7kgd source: https://lightmysky.com/learn/mathematics/the-definite-integral-and-the-area-under-a-curve-mt_S2IO2S7kgd.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # The Definite Integral and the Area Under a Curve Estimate the area under a curve with rectangles, then evaluate it exactly by substituting the limits into the antiderivative. Areas below the axis come out negative and have to be handled separately. Subject: Mathematics · Area: Calculus & Analysis · Ages 17 to 18 Page: https://lightmysky.com/learn/mathematics/the-definite-integral-and-the-area-under-a-curve-mt_S2IO2S7kgd ## Ready when they can - Estimate an area with rectangles and compare it with the exact value - Evaluate a definite integral using square-bracket notation, or combine supplied integral values using interval additivity and constant scaling - Split an integral at a root so that area below the axis is counted correctly ## Lesson: From rectangles to exact area For a continuous function, estimate its integral on [a, b] with equal-width rectangles. Width is (b - a)/n. Multiply each sampled function value by that width, then add. Use left endpoints, right endpoints or midpoints. For an increasing nonnegative function, left sums underestimate area and right sums overestimate; decreasing functions reverse this. As the widths tend to zero, the sums approach the exact integral. **Example.** If F' = f on [a, b], evaluate ∫ from a to b of f(x) dx as [F(x)] from a to b = F(b) - F(a). For 3x² from 1 to 2, this gives [x³] = 8 - 1 = 7. Any additive constant cancels. When f is nonnegative, this integral equals geometric area. A definite integral is a number; an indefinite integral names a family. An integral counts above-axis regions positively and below-axis regions negatively. For total geometric area, split at roots within the interval, check the sign on each stretch and add the absolute values of the integrals. For y = x - 2 on [0, 3], [x²/2 - 2x] gives -2 on [0, 2] and 1/2 on [2, 3]. The signed total is -3/2; the area is 2 + 1/2 = 5/2. *(drawing: For area, take each region's absolute value before adding.)* You can use given integral values without knowing a formula for f. Adjacent intervals add: ∫ from a to c of f = ∫ from a to b of f + ∫ from b to c of f. A constant multiplier stays outside: ∫ of kf = k∫ of f. If the parts are 4 and -1, the whole is 3 and twice the whole is 6. A zero signed total can still contain positive area when opposite contributions cancel. *(drawing: Keep each contribution's sign when finding the integral.)* **Recap.** Rectangles estimate, brackets give the exact value, and roots tell you where to split so below-axis parts count positive. *Lesson and practice text written for this dataset.* ## Practice 15 questions on this page, each with its working shown. ## Needs first - [Integration as the Reverse of Differentiation](https://lightmysky.com/learn/mathematics/integration-as-the-reverse-of-differentiation-mt_edtk3ArxRk) ## Optional connections - [Sketching a Curve from Its Factorised Form](https://lightmysky.com/learn/mathematics/sketching-a-curve-from-its-factorised-form-mt_x4tZLdH8fh) ## Opens up - [The Electric Field of a Continuous Charge Distribution](https://lightmysky.com/learn/science/the-electric-field-of-a-continuous-charge-distribution-mt_IIcVqWapC1) - [Areas Between Two Curves](https://lightmysky.com/learn/mathematics/areas-between-two-curves-mt_lDBZmnLK1b) - [Riemann Sums and the Definite Integral as a Limit](https://lightmysky.com/learn/mathematics/riemann-sums-and-the-definite-integral-as-a-limit-mt_pwa2SRU6P9) - [The Trapezium Rule](https://lightmysky.com/learn/mathematics/the-trapezium-rule-mt_rAlJ1WIRmZ) - [Work as a Line Integral Along a Path](https://lightmysky.com/learn/science/work-as-a-line-integral-along-a-path-mt_UFuQUPGAt7)