A definite integral measures the net accumulated quantity represented by a function over an interval. When f(x) is positive, the integral ∫_a^b f(x) dx is the area under the curve from x = a to x = b. In many real problems, the exact antiderivative is hard to find or the data come only from measurements.
Estimation methods let us approximate the integral using rectangles, trapezoids, or bounds from known function behavior.
The basic idea is to divide [a,b] into small subintervals and add simple areas that approximate the curved region. Left, right, midpoint, and trapezoidal sums use different sample points or shapes, so they can overestimate or underestimate depending on how the function changes. Comparison, maximum, and minimum properties help place guaranteed bounds on the integral without computing it exactly.
Error bounds connect the smoothness of the curve to how accurate an estimate must be.
Key Facts
- Definite integral as area for f(x) ≥ 0: ∫_a^b f(x) dx = area under y = f(x) from a to b.
- Equal subinterval width: Δx = (b - a)/n.
- Riemann sum estimate: ∫_a^b f(x) dx ≈ Σ f(x_i*) Δx, where x_i* is a chosen sample point in subinterval i.
- If m ≤ f(x) ≤ M on [a,b], then m(b - a) ≤ ∫_a^b f(x) dx ≤ M(b - a).
- If f(x) ≤ g(x) on [a,b], then ∫_a^b f(x) dx ≤ ∫_a^b g(x) dx.
- Trapezoidal error bound: |E_T| ≤ K(b - a)^3/(12n^2) if |f''(x)| ≤ K on [a,b].
Vocabulary
- Definite integral
- A definite integral is the accumulated signed area represented by a function over a fixed interval.
- Riemann sum
- A Riemann sum estimates an integral by adding areas of rectangles whose heights come from function values.
- Partition
- A partition is a division of an interval into smaller subintervals used for approximation.
- Trapezoidal sum
- A trapezoidal sum estimates an integral by replacing pieces of the curve with line segments and adding trapezoid areas.
- Error bound
- An error bound is a guaranteed maximum possible difference between an estimate and the exact value.
Common Mistakes to Avoid
- Forgetting Δx in a Riemann sum is wrong because the sum of heights alone does not have units of area or accumulation.
- Assuming every right sum is an overestimate is wrong because right sums overestimate only for increasing positive functions and underestimate for decreasing positive functions.
- Using the maximum value as the exact integral is wrong because M(b - a) is only an upper bound when f(x) ≤ M on the whole interval.
- Applying an error bound without checking its conditions is wrong because formulas such as |E_T| ≤ K(b - a)^3/(12n^2) require a valid bound on |f''(x)| over the interval.
Practice Questions
- 1 Estimate ∫_0^4 (x^2 + 1) dx using a left Riemann sum with n = 4 equal subintervals.
- 2 Suppose 2 ≤ f(x) ≤ 7 for all x in [1,5]. Find lower and upper bounds for ∫_1^5 f(x) dx.
- 3 A positive function is increasing and concave up on [a,b]. Explain whether the left Riemann sum, right Riemann sum, and trapezoidal sum tend to be underestimates or overestimates of ∫_a^b f(x) dx.