Respuesta :
Answer:
Average value of the function over the interval is 3.
Step-by-step explanation:
Average value of the function y = 2x³ - 3x² over the interval (-1, 3) will be defined by the area under curve for the given interval divided by width of the interval.
A = [tex]\int\limits^3_ {-1} \, (2x^{3}-3x^{2})dx[/tex]
= [tex][\frac{2x^{4}}{4}-\frac{3x^{3}}{3}]^{3}_{(-1)}[/tex]
= [tex][\frac{x^{4}}{2}]^{3}_{-1}-[x^{3}]^{3}_{-1}[/tex]
= [tex][\frac{3^{4}-(-1)^{4} }{2}]-[3^{3}-(-1)^{3}][/tex]
= [tex][\frac{81-1}{2}]-[27+1][/tex]
= [tex]40-28[/tex]
= 12
Width of the interval = 3 - (-1) = 4
Average value = [tex]\frac{12}{4}=3[/tex]
Therefore, average value of the function over the given interval is 3.
