5.3 Definite Integralsand AntiderivativesOrgan Pipe Cactus National Monument, ArizonaGreg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2009
Your textbook gives rules for working with integrals, the most important of which are:2.0aafx dxIf the upper and lower limits are equal, then the integral is zero.1.baabfx dxfx dxReversing the limits changes the sign.bbaakfx dxkfx dx3.Constant multiples can be moved outside.
1.0aafx dxIf the upper and lower limits are equal, then the integral is zero.2.baabfx dxfx dxReversing the limits changes the sign.bbaakfx dxkfx dx3.Constant multiples can be moved outside.bbbaaafxg xdxfx dxg x dx4.Integrals can be added and subtracted.
bbbaaafxg xdxfx dxg x dx4.Integrals can be added and subtracted.5.bccabafx dxfx dxfx dxIntervalscan be added(or subtracted.)abcyfx
The average valueof a function is the value that would give the same area if the function was a constant:212yx3212Ax dx3316x276924.54.5Average Value1.53Area1Average ValueWidthbafxdxba1.5
The mean value theorem for definite integrals says that for a continuous function, at some point on the interval the actual value will equal to the average value.Mean Value Theorem (for definite integrals)If fis continuous on then at some point c in , ,a b,a b1bafcfxdxbaNote: When a problem on the AP Exam refers to the “Mean Value Theorem” without specifying which one, they mean the Mean Value Theorem for Derivatives.