Υπολογιστής Ορισμένου και Ακατάλληλου Ολοκληρώματος

Υπολογίστε ορισμένα και ακατάλληλα ολοκληρώματα βήμα προς βήμα

Η αριθμομηχανή θα προσπαθήσει να υπολογίσει το ορισμένο (δηλ. με όρια) ολοκλήρωμα, συμπεριλαμβανομένου του ατελούς, παρουσιάζοντας τα βήματα.

Enter a function:

Integrate with respect to:

Enter a lower limit:

If you need `-oo`, type -inf.

Enter an upper limit:

If you need `oo`, type inf.

Please write without any differentials such as `dx`, `dy` etc.

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Solution

Your input: calculate $$$\int_{\sqrt{6}}^{4}\left( 6 - x^{2} \right)dx$$$

First, calculate the corresponding indefinite integral: $$$\int{\left(6 - x^{2}\right)d x}=\frac{x \left(18 - x^{2}\right)}{3}$$$ (for steps, see indefinite integral calculator)

According to the Fundamental Theorem of Calculus, $$$\int_a^b F(x) dx=f(b)-f(a)$$$, so just evaluate the integral at the endpoints, and that's the answer.

$$$\left(\frac{x \left(18 - x^{2}\right)}{3}\right)|_{\left(x=4\right)}=\frac{8}{3}$$$

$$$\left(\frac{x \left(18 - x^{2}\right)}{3}\right)|_{\left(x=\sqrt{6}\right)}=4 \sqrt{6}$$$

$$$\int_{\sqrt{6}}^{4}\left( 6 - x^{2} \right)dx=\left(\frac{x \left(18 - x^{2}\right)}{3}\right)|_{\left(x=4\right)}-\left(\frac{x \left(18 - x^{2}\right)}{3}\right)|_{\left(x=\sqrt{6}\right)}=\frac{8}{3} - 4 \sqrt{6}$$$

Answer: $$$\int_{\sqrt{6}}^{4}\left( 6 - x^{2} \right)dx=\frac{8}{3} - 4 \sqrt{6}\approx -7.13129230446605$$$