Antiderivatives and Indefinite Integrals The function F(x) is the antiderivative of f(x) if: Notation: How many antiderivatives of are there? Why? Verify the antiderivative: What is the power rule for an antiderivative? Find: Finding a particular solution: if F(1) = 3 1
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Antiderivatives and Indefinite IntegralsThe function F(x) is the antiderivative of f(x) if: Notation:
How many antiderivatives of are there? Why?
Verify the antiderivative:
What is the power rule for an antiderivative?
Find: Finding a particular solution:
if F(1) = 3
1
Whiteboard warm up - Anti Derivatives:
Find the particular solution to if F(2) = 5
Test Yourself: Find the antiderivative
x2−2x+5− 2x sin x−e x
√ x ( x+1 )x ( x−5 )3√x
Definite Integrals
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Estimate the area under the curve on the interval [–2,1], using 3 rectangles:
Upper limit: Lower Limit:
Theorem 4.3: Limit of Upper and Lower Sums
https://www.desmos.com/calculator/tgyr42ezjqΔx=
Right: x i=a+iΔx
Left: x i=a+(i−1 ) Δx
Using limits, find the area under the curve on the interval [–2,1]
Find the antiderivative of:So far the examples we’ve used have all been carefully chosen to have a nice antiderivative. In practice there are lots of functions whose antiderivative is not so simple.Riemann Sums:
Let’s consider Approximate the definite integral using the left and right point rule with 4 equally spaced intervals
Whiteboards:Approximate the definite integral using the midpoint point rule with 4 equally spaced intervals
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Area of a trapezoid
Use the trapezoid rule to approximate, for n = 4 (use calculator)
Whiteboards:
f is a continuous function for which and on the interval [3,10]. Use the values in the table below
to estimate the value of with the trapezoid, left, and right point rules. Are the values an over/under estimate? Justify your answer.
Suppose a function has the following properties on the interval [1,5]: f ' ( x )>0 and f '' ( x )>0Form an inequality comparing an estimate made by the midpoint (M), trapezoid (T), right (R), and left (L) with