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.
What a learner can do afterwards
- 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
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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.
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.
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.
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.
Readings, additional questions and flashcards written for this dataset. Referenced videos are by Khan Academy.
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