Definite Integral

Definite Integral

For all of the following, suppose that $\displaystyle k$, $\displaystyle a$, $\displaystyle b$, and $\displaystyle c$ are constants with $\displaystyle a < b < c$, and that $\displaystyle f$ and $\displaystyle g$ are integrable functions on the domain of integration.

THEOREM

Definite Integral

$\displaystyle \begin{aligned}\int_a^b k \ dx = (b - a)k\end{aligned}$

THEOREM

Definite Integral

$\displaystyle \begin{aligned}\int_a^a f(x) \ dx = 0\end{aligned}$

THEOREM

Definite Integral

$\displaystyle \begin{aligned}\int_b^a f(x) \ dx = -\int_a^b f(x) \ dx\end{aligned}$

THEOREM

Definite Integral

$\displaystyle \begin{aligned}\int_a^c f(x) \ dx = \int_a^b f(x) \ dx + \int_b^c f(x) \ dx\end{aligned}$

THEOREM

Definite Integral

$\displaystyle \begin{aligned}\int_a^b kf(x) \ dx = k \int_a^b f(x) \ dx\end{aligned}$

THEOREM

Definite Integral

$\displaystyle \begin{aligned}\int_a^b (f(x) \pm g(x)) \ dx = \int_a^b f(x) \ dx \pm \int_a^b g(x) \ dx\end{aligned}$

HANDS-ON PRACTICE

Evaluate definite integrals

Medium·20 minShow reference answer

Evaluate.

(a)

$\displaystyle \int_1^2 {1 \over t} \ dt$

(b)

$\displaystyle \int_{-2}^1 x^3 \ dx$

(c)

$\displaystyle \int_1^5 s(s^2 + 1) \ ds$

(d)

$\displaystyle \int_{\sqrt{\ln(2)}}^{\sqrt{\ln(4)}} xe^{x^2} \ dx$

(e)

$\displaystyle \int_x^{x^2} \sin(t) \ dt$

(f)

$\displaystyle \int_5^{13} (x + 1)\sqrt{2x - 1} \ dx$

INTERACTIVE LAB

Move the limits and change the integral’s sign

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Adjust the limits and constant multiplier. Make the endpoints coincide, reverse their order, or choose a negative k; compare the signed integral with the shaded area.

Lower limit a

1

Upper limit b

4

k

1

$\displaystyle \int_{1}^{4} 1f(x)\,dx=6.596973$

Geometric area of the shading: 6.596973

KNOWLEDGE QUIZ

Retry

QUESTION

If $\displaystyle \int_a^b f(x)\,dx=3$, what is $\displaystyle \int_b^a f(x)\,dx$?

A$\displaystyle 0$B$\displaystyle 6$C$\displaystyle -3$D$\displaystyle 3$