The definite integral and its properties

Three quick questions from earlier lessons.

$\displaystyle\int_0^2 (-3)\,dx$ : the graph $y = -3$ lies below the axis. What is the integral?

The definite integral is the limit of Riemann sums: a signed area.

\displaystyle\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum f(x_i^*)\Delta x

. Properties:

\displaystyle\int_b^a f = -\int_a^b f

;

\displaystyle\int_a^c f = \int_a^b f + \int_b^c f

;

\displaystyle\int_a^b (kf + g) = k\int_a^b f + \int_a^b g

;

\displaystyle\int_a^b c\,dx = c(b - a)

.

$\displaystyle\int_0^4 f = 5$ and $\displaystyle\int_4^6 f = -2$ . Find $\displaystyle\int_0^6 \big(2f(x) - 1\big)\,dx$ .

\displaystyle\int_0^6 f = 5 + (-2) = 3

Join the intervals.

2 \cdot 3 - 1 \cdot 6

Linearity; the constant −1 over a length of 6.

= 0

Compute.

Your turn: with the same $f$ , find $\displaystyle\int_6^0 f(x)\,dx$ .

-3

Reverse the limits: change the sign.

$\displaystyle\int_1^3 f(x)\,dx = 7$ and $\displaystyle\int_3^7 f(x)\,dx = -4$ . Find $\displaystyle\int_7^1 f(x)\,dx$ .
$\displaystyle\int_1^3 f(x)\,dx = -7$ and $\displaystyle\int_3^7 f(x)\,dx = 8$ . Find $\displaystyle\int_7^1 f(x)\,dx$ .

$\displaystyle\int_0^{ 4 } f(x)\,dx = 3$ . Find $\displaystyle\int_0^{ 4 } \big(3f(x) + 2\big)\,dx$ .
$\displaystyle\int_0^{ 9 } f(x)\,dx = 4$ . Find $\displaystyle\int_0^{ 9 } \big(3f(x) + 2\big)\,dx$ .

Three questions to check you have got it.

$\displaystyle\int_1^3 f(x)\,dx = 7$ and $\displaystyle\int_3^7 f(x)\,dx = -4$ . Find $\displaystyle\int_7^1 f(x)\,dx$ .
$\displaystyle\int_1^3 f(x)\,dx = -7$ and $\displaystyle\int_3^7 f(x)\,dx = 8$ . Find $\displaystyle\int_7^1 f(x)\,dx$ .
$\displaystyle\int_1^3 f(x)\,dx = 5$ and $\displaystyle\int_3^7 f(x)\,dx = 7$ . Find $\displaystyle\int_7^1 f(x)\,dx$ .

The definite integral

Watch out: a constant is integrated over the whole length.

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