09 Definite Integrals and the Fundamental Theorem Free Online FlashCards

What is a Riemann sum?

A Riemann sum is Σ f(xᵢ*)Δx, adding signed rectangle areas over subintervals to approximate accumulation.

How is Δx defined for n equal subintervals?

The equal subinterval width is Δx = (b − a)/n, where n is the number of subintervals.

What does the variable of integration represent?

In a definite integral, the variable of integration is a placeholder: ∫ₐᵇ f(x) dx = ∫ₐᵇ f(t) dt.

What does a definite integral measure geometrically?

A definite integral gives net signed area: area above the axis minus area below it. For areas 7 and 3, the value is 4.

How is total geometric area found?

Total geometric area is ∫ₐᵇ |f(x)| dx. Split at sign changes or use the absolute value so regions do not cancel.

What happens when the limits of integration are reversed?

Reversing the limits changes the sign: ∫ᵇₐ f(x) dx = −∫ₐᵇ f(x) dx.

What does integrating a rate of change produce?

The accumulation principle states Q(b) − Q(a) = ∫ₐᵇ r(t) dt: integrating a rate gives the quantity’s net change.

For v(t) = t − 2 on [0, 4], what are displacement and distance?

Displacement is ∫₀⁴(t − 2) dt = 0, but total distance is ∫₀⁴|t − 2| dt = 4 units because velocity changes sign.

What does the Fundamental Theorem of Calculus, Part 1, state?

If F(x) = ∫ₐˣ f(t) dt and f is continuous, then F′(x) = f(x). Differentiation reverses accumulation.

Find G′(x) if G(x) = ∫₂ˣ³ cos(t²) dt.

G′(x) = 3x² cos(x⁶). Apply the Fundamental Theorem to the integrand, then multiply by the derivative of x³.

What does the Fundamental Theorem of Calculus, Part 2, state?

If F′(x) = f(x), then ∫ₐᵇ f(x) dx = F(b) − F(a). This is the evaluation theorem.

What does [F(x)]ₐᵇ mean?

The notation [F(x)]ₐᵇ means F(b) − F(a), or upper endpoint value minus lower endpoint value.