Definite Integrals

What is a definite integral?

A definite integral $\int_{a}^{b} f(x)\, dx$ gives the signed area under the curve $f(x)$ between $x = a$ and $x = b$ . By the fundamental theorem of calculus, if $F(x)$ is an antiderivative of $f(x)$ (that is, $F'(x) = f(x)$ ), then

\int_{a}^{b} f(x)\, dx = F(b) - F(a)

Worked example

Evaluate $\displaystyle\int_{0}^{\pi/2} \cos(2x)\, dx$ .

Solution: An antiderivative of $\cos(2x)$ is $\dfrac{\sin(2x)}{2}$ . Applying the limits:

\left[\dfrac{\sin(2x)}{2}\right]_{0}^{\pi/2} = \dfrac{\sin(\pi)}{2} - \dfrac{\sin(0)}{2} = \dfrac{0}{2} - \dfrac{0}{2} = 0

Definite Integrals

What is a definite integral?

\int_{a}^{b} f(x)\, dx = F(b) - F(a)

Worked example

Evaluate $\displaystyle\int_{0}^{\pi/2} \cos(2x)\, dx$ .

Solution: An antiderivative of $\cos(2x)$ is $\dfrac{\sin(2x)}{2}$ . Applying the limits:

\left[\dfrac{\sin(2x)}{2}\right]_{0}^{\pi/2} = \dfrac{\sin(\pi)}{2} - \dfrac{\sin(0)}{2} = \dfrac{0}{2} - \dfrac{0}{2} = 0

Definite Integrals

What is a definite integral?

Worked example

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Definite Integrals

What is a definite integral?

Worked example

Continue learning