问题标题:
急问(高等微积分)Letaandbbepositivenumbers.Findthevalueof∫[ax+b]dxinthefollowingtwoways:a.Usingelementarygeometry,interpreting∫[ax+b]dxasanarea.b.UsingtheFirstFundamentalTheorem(IntegratingDerivatives).注:上下
更新时间:2024-04-28 07:02:19
问题描述:
急问(高等微积分)
Letaandbbepositivenumbers.Findthevalueof∫[ax+b]dxinthefollowingtwoways:
a.Usingelementarygeometry,interpreting∫[ax+b]dxasanarea.
b.UsingtheFirstFundamentalTheorem(IntegratingDerivatives).
注:上下界皆为1,0
宋臣回答:
a)使用面积原理,直线在(0,1)下的图形为梯形,面积为
∫[ax+b]dx=(1/2)*(b+a+b)*1=(1/2)(a+2b)
b)积分
∫[ax+b]dx=(1/2)ax^2+bx在0到1上计算=(1/2)a+b=(1/2)(a+2b)
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