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tino4ka555 [31]
3 years ago
7

How to evaluate definite integral without the fundamental theorem of calculus?

Mathematics
1 answer:
AURORKA [14]3 years ago
7 0
If we can find the antiderivative function
F
(
x
)
of the integrand
f
(
x
)
, then the definite integral
∫
b
a
f
(
x
)
d
x
can be determined by
F
(
b
)
−
F
(
a
)
provided that
f
(
x
)
is continuous.
We are usually given continuous functions, but if you want to be rigorous in your solutions, you should state that
f
(
x
)
is continuous and why.
FTC part 2 is a very powerful statement. Recall in the previous chapters, the definite integral was calculated from areas under the curve using Riemann sums. FTC part 2 just throws that all away. We just have to find the antiderivative and evaluate at the bounds! This is a lot less work.
For most students, the proof does give any intuition of why this works or is true. But let's look at
s
(
t
)
=
∫
b
a
v
(
t
)
d
t
. We know that integrating the velocity function gives us a position function. So taking
s
(
b
)
−
s
(
a
)
results in a displacement.
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Answer:

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Step-by-step explanation:

1 Subtract z from both sides.

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Step-by-step explanation:

1 Subtract h from both sides

-\frac{4}{j} =k-h

2 Multiply both sides by j.

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3 Divide both sides by k−h.

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