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 $$$\int_{0}^{2}\left( y e^{2} \right)dy$$$

First, calculate the corresponding indefinite integral: $$$\int{y e^{2} d y}=\frac{y^{2} e^{2}}{2}$$$ (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.

$$$\left(\frac{y^{2} e^{2}}{2}\right)|_{\left(y=2\right)}=2 e^{2}$$$

$$$\left(\frac{y^{2} e^{2}}{2}\right)|_{\left(y=0\right)}=0$$$

$$$\int_{0}^{2}\left( y e^{2} \right)dy=\left(\frac{y^{2} e^{2}}{2}\right)|_{\left(y=2\right)}-\left(\frac{y^{2} e^{2}}{2}\right)|_{\left(y=0\right)}=2 e^{2}$$$

Answer: $$$\int_{0}^{2}\left( y e^{2} \right)dy=2 e^{2}\approx 14.7781121978613$$$

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