Further Definite Integrals

Further Definite Integrals

Conceptual Understanding

• Definite Integral represents the accumulation of quantities, such as areas under curves or total distance travelled.
• Fundamental Theorem of Calculus, the integral of a function over an interval is equal to the antiderivative evaluated at the end points.

Calculation Methods

• Definite integrals are typically computed using the Fundamental Theorem of Calculus. The process usually involves finding an antiderivative of the function and then using the Fundamental Theorem to evaluate the integral.
• To solve improper integrals (integrals with infinite limits or discontinuous integrands), apply the limit concept.

Properties of Definite Integrals

• Symmetry Properties, if f(x) is an even function then the integral from -a to a is twice the integral from 0 to a.
• Definite integrals have linearity property, meaning the integral of a sum of functions is equal to the sum of the integrals.
• If a function is non-negative over [a, b], then its definite integral over [a, b] is also non-negative.
• The integral of the function from a to b is equal to the negative of the integral from b to a.

Applications of Definite Integrals

• Calculation of areas under curves and volumes of solids by revolution.
• In Physics, definite integrals are used to calculate total distance travelled, work done, or charge in a circuit.
• In Statistics, definite integrals calculate total probabilities under a probability distribution curve.