If we use this type of partition(subdivision we have a Riemann Sum n s i =1 f c i)\u0394 x i where x i-1 \u2264 c i \u2264 x i In order for a Riemann Sum to

If we use this type of partition(subdivision we have

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7. Iffgon [a, b], thenintegraldisplaybaf(x)dxintegraldisplaybag(x)dxExample: Given thatintegraldisplay1f(x)dx= 0 andintegraldisplay1f(x)dx= 5, andintegraldisplay4f(x)dx= 9 , find 4
5.4 The Fundamental Theorem of CalculusObjectives1. Evaluate a definite integral using the Fundamental Theorem of Calculus (FTC)2. Understand and use the Mean Value Theorem for integrals3. Find the average value of a function over a closed interval4. Understand and use the Second Fundamental Theorem of CalculusNow, even though the definite integral gives us the (net) area between a curve and thex-axis, westill want a better way to evaluate it than as a limit of a summation. This is where the FundamentalTheorem of Calculus (FTC for short) comes in:Fundamental Theorem of Calculus:Iffis continuous on the closed interval [a, b] andFisan antiderivative offon [a, b], thenintegraldisplaybaf(x)dx=F(b)-F

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