undefined To use the Fundamental Theorem of Calculus to evaluate ∫ f(x) dx from a to b: 1. Find an antiderivative, F(x), of the function f(x). 2. Evaluate F(b) - F(a). This gives the value of the definite integral ∫ f(x) dx from a to b.

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How to use the Fundamental Theorem of Calculus to evaluate ?

Answer 1

Emilia Aguilar

#int_(-pi)^pif(x)dx=3pi^2-9#

The Fundamental Theorem of Calculus states that #int_a^bf(x)dx=[g(x)]_a^b=g(b)-g(a)#, where #g(x)# is the antiderivative of #f(x)#, ie #g'(x)=f(x)#.

Since the given function #f(x)# is a compound function with different definitions each side of #0#, we have to split the integral into #2# parts and select the limits of integration accordingly, that is,

#int_(-pi)^pif(x)dx=int_(-pi)^0 6xdx+int_0^pi-9sinxdx#

#=6[x^2/2]_(-pi)^0+[9cosx]_0^pi#

#=6(0-pi^2/2)+9(cospi-cos0)#

#=3pi^2-9#

#=20.6088#

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Answer 2

Emily Ammerman

Split the integral into two pieces.

#int_-pi^pi f(x) dx = int_-pi^0 f(x) dx + int_0^pi f(x) dx#

# = int_-pi^0 6x dx + int_0^pi -9sin(x) dx#

# = 3x^2]_-pi^0 + 9 cosx]_0^pi#

# = [3(0)^2-3(-pi)^2] + [9cos(pi)-9cos(0)]#

# = -3pi^2-18#

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Answer 3

Evelyn Agard

To use the Fundamental Theorem of Calculus to evaluate ∫ f(x) dx from a to b:

Find an antiderivative, F(x), of the function f(x).
Evaluate F(b) - F(a). This gives the value of the definite integral ∫ f(x) dx from a to b.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.