Integration and the Fundamental Theorem Free Online FlashCards

What is an antiderivative?

An antiderivative F of f satisfies F′(x) = f(x) throughout the interval.

Why does an indefinite integral include + C?

The constant C represents the arbitrary constant shared by all antiderivatives of a function.

State the power rule for integration.

∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C, provided n ≠ −1.

What is integration by substitution?

Substitution reverses the chain rule by setting u = g(x), replacing du with g′(x) dx, integrating in u, and then substituting back.

How is the trapezoidal rule related to endpoint sums?

The trapezoidal estimate is Tₙ = (Lₙ + Rₙ)/2, equivalently averaging the left and right Riemann sums.

How is a definite integral defined?

A definite integral is the limit of Riemann sums as the subinterval width approaches zero, when that limit exists.

What does Fundamental Theorem Part 1 state?

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

How do velocity integrals distinguish displacement from distance?

Displacement is ∫ₐᵇ v(t) dt, whereas total distance is ∫ₐᵇ |v(t)| dt.

What is the integral of 1/x?

∫ 1/x dx = ln|x| + C. The absolute value works for both positive and negative x ≠ 0.

Evaluate ∫ cos(3x) dx.

∫ cos(3x) dx = (1/3)sin(3x) + C. The factor 1/3 compensates for the derivative of 3x.

How do endpoint sums behave for an increasing function?

For an increasing function, the left Riemann sum usually underestimates and the right Riemann sum usually overestimates the integral.

How do total area and signed area differ?

∫ₐᵇ |f(x)| dx gives total geometric area, while ∫ₐᵇ f(x) dx gives net signed accumulation.